:: RELAT_1 semantic presentation
:: deftheorem Def1 defines Relation-like RELAT_1:def 1 :
for b
1 being
set holds
( b
1 is
Relation-like iff for b
2 being
set holds
not ( b
2 in b
1 & ( for b
3, b
4 being
set holds
not b
2 = [b3,b4] ) ) );
theorem Th1: :: RELAT_1:1
canceled;
theorem Th2: :: RELAT_1:2
canceled;
theorem Th3: :: RELAT_1:3
theorem Th4: :: RELAT_1:4
theorem Th5: :: RELAT_1:5
theorem Th6: :: RELAT_1:6
Lemma4:
for b1, b2 being set
for b3 being Relation holds
( [b1,b2] in b3 implies ( b1 in union (union b3) & b2 in union (union b3) ) )
:: deftheorem Def2 defines = RELAT_1:def 2 :
for b
1, b
2 being
Relation holds
( b
1 = b
2 iff for b
3, b
4 being
set holds
(
[b3,b4] in b
1 iff
[b3,b4] in b
2 ) );
:: deftheorem Def3 defines c= RELAT_1:def 3 :
for b
1, b
2 being
Relation holds
( b
1 c= b
2 iff for b
3, b
4 being
set holds
(
[b3,b4] in b
1 implies
[b3,b4] in b
2 ) );
theorem Th7: :: RELAT_1:7
canceled;
theorem Th8: :: RELAT_1:8
canceled;
theorem Th9: :: RELAT_1:9
theorem Th10: :: RELAT_1:10
:: deftheorem Def4 defines dom RELAT_1:def 4 :
for b
1 being
Relationfor b
2 being
set holds
( b
2 = dom b
1 iff for b
3 being
set holds
( b
3 in b
2 iff ex b
4 being
set st
[b3,b4] in b
1 ) );
theorem Th11: :: RELAT_1:11
canceled;
theorem Th12: :: RELAT_1:12
canceled;
theorem Th13: :: RELAT_1:13
theorem Th14: :: RELAT_1:14
theorem Th15: :: RELAT_1:15
:: deftheorem Def5 defines rng RELAT_1:def 5 :
for b
1 being
Relationfor b
2 being
set holds
( b
2 = rng b
1 iff for b
3 being
set holds
( b
3 in b
2 iff ex b
4 being
set st
[b4,b3] in b
1 ) );
theorem Th16: :: RELAT_1:16
canceled;
theorem Th17: :: RELAT_1:17
canceled;
theorem Th18: :: RELAT_1:18
theorem Th19: :: RELAT_1:19
theorem Th20: :: RELAT_1:20
theorem Th21: :: RELAT_1:21
theorem Th22: :: RELAT_1:22
theorem Th23: :: RELAT_1:23
theorem Th24: :: RELAT_1:24
theorem Th25: :: RELAT_1:25
theorem Th26: :: RELAT_1:26
theorem Th27: :: RELAT_1:27
theorem Th28: :: RELAT_1:28
:: deftheorem Def6 defines field RELAT_1:def 6 :
theorem Th29: :: RELAT_1:29
theorem Th30: :: RELAT_1:30
theorem Th31: :: RELAT_1:31
theorem Th32: :: RELAT_1:32
theorem Th33: :: RELAT_1:33
theorem Th34: :: RELAT_1:34
definition
let c
1 be
Relation;
func c
1 ~ -> Relation means :
Def7:
:: RELAT_1:def 7
for b
1, b
2 being
set holds
(
[b1,b2] in a
2 iff
[b2,b1] in a
1 );
existence
ex b1 being Relation st
for b2, b3 being set holds
( [b2,b3] in b1 iff [b3,b2] in c1 )
uniqueness
for b1, b2 being Relation holds
( ( for b3, b4 being set holds
( [b3,b4] in b1 iff [b4,b3] in c1 ) ) & ( for b3, b4 being set holds
( [b3,b4] in b2 iff [b4,b3] in c1 ) ) implies b1 = b2 )
involutiveness
for b1, b2 being Relation holds
( ( for b3, b4 being set holds
( [b3,b4] in b1 iff [b4,b3] in b2 ) ) implies for b3, b4 being set holds
( [b3,b4] in b2 iff [b4,b3] in b1 ) )
;
end;
:: deftheorem Def7 defines ~ RELAT_1:def 7 :
for b
1, b
2 being
Relation holds
( b
2 = b
1 ~ iff for b
3, b
4 being
set holds
(
[b3,b4] in b
2 iff
[b4,b3] in b
1 ) );
theorem Th35: :: RELAT_1:35
canceled;
theorem Th36: :: RELAT_1:36
canceled;
theorem Th37: :: RELAT_1:37
theorem Th38: :: RELAT_1:38
theorem Th39: :: RELAT_1:39
theorem Th40: :: RELAT_1:40
theorem Th41: :: RELAT_1:41
definition
let c
1, c
2 be
Relation;
func c
1 * c
2 -> Relation means :
Def8:
:: RELAT_1:def 8
for b
1, b
2 being
set holds
(
[b1,b2] in a
3 iff ex b
3 being
set st
(
[b1,b3] in a
1 &
[b3,b2] in a
2 ) );
existence
ex b1 being Relation st
for b2, b3 being set holds
( [b2,b3] in b1 iff ex b4 being set st
( [b2,b4] in c1 & [b4,b3] in c2 ) )
uniqueness
for b1, b2 being Relation holds
( ( for b3, b4 being set holds
( [b3,b4] in b1 iff ex b5 being set st
( [b3,b5] in c1 & [b5,b4] in c2 ) ) ) & ( for b3, b4 being set holds
( [b3,b4] in b2 iff ex b5 being set st
( [b3,b5] in c1 & [b5,b4] in c2 ) ) ) implies b1 = b2 )
end;
:: deftheorem Def8 defines * RELAT_1:def 8 :
for b
1, b
2, b
3 being
Relation holds
( b
3 = b
1 * b
2 iff for b
4, b
5 being
set holds
(
[b4,b5] in b
3 iff ex b
6 being
set st
(
[b4,b6] in b
1 &
[b6,b5] in b
2 ) ) );
theorem Th42: :: RELAT_1:42
canceled;
theorem Th43: :: RELAT_1:43
canceled;
theorem Th44: :: RELAT_1:44
theorem Th45: :: RELAT_1:45
theorem Th46: :: RELAT_1:46
theorem Th47: :: RELAT_1:47
theorem Th48: :: RELAT_1:48
for b
1, b
2, b
3 being
Relation holds
( b
1 c= b
2 implies b
3 * b
1 c= b
3 * b
2 )
theorem Th49: :: RELAT_1:49
for b
1, b
2, b
3 being
Relation holds
( b
1 c= b
2 implies b
1 * b
3 c= b
2 * b
3 )
theorem Th50: :: RELAT_1:50
for b
1, b
2, b
3, b
4 being
Relation holds
( b
1 c= b
2 & b
3 c= b
4 implies b
1 * b
3 c= b
2 * b
4 )
theorem Th51: :: RELAT_1:51
theorem Th52: :: RELAT_1:52
theorem Th53: :: RELAT_1:53
for b
1, b
2, b
3 being
Relation holds
(b1 * b2) \ (b1 * b3) c= b
1 * (b2 \ b3)
theorem Th54: :: RELAT_1:54
theorem Th55: :: RELAT_1:55
for b
1, b
2, b
3 being
Relation holds
(b1 * b2) * b
3 = b
1 * (b2 * b3)
theorem Th56: :: RELAT_1:56
for b
1 being
Relation holds
( ( for b
2, b
3 being
set holds
not
[b2,b3] in b
1 ) implies b
1 = {} )
theorem Th57: :: RELAT_1:57
canceled;
theorem Th58: :: RELAT_1:58
canceled;
theorem Th59: :: RELAT_1:59
canceled;
theorem Th60: :: RELAT_1:60
theorem Th61: :: RELAT_1:61
canceled;
theorem Th62: :: RELAT_1:62
theorem Th63: :: RELAT_1:63
theorem Th64: :: RELAT_1:64
theorem Th65: :: RELAT_1:65
theorem Th66: :: RELAT_1:66
theorem Th67: :: RELAT_1:67
:: deftheorem Def9 defines non-empty RELAT_1:def 9 :
:: deftheorem Def10 defines id RELAT_1:def 10 :
for b
1 being
set for b
2 being
Relation holds
( b
2 = id b
1 iff for b
3, b
4 being
set holds
(
[b3,b4] in b
2 iff ( b
3 in b
1 & b
3 = b
4 ) ) );
theorem Th68: :: RELAT_1:68
canceled;
theorem Th69: :: RELAT_1:69
canceled;
theorem Th70: :: RELAT_1:70
canceled;
theorem Th71: :: RELAT_1:71
theorem Th72: :: RELAT_1:72
theorem Th73: :: RELAT_1:73
for b
1 being
set for b
2 being
Relation holds
( ( for b
3 being
set holds
( b
3 in b
1 implies
[b3,b3] in b
2 ) ) implies
id b
1 c= b
2 )
theorem Th74: :: RELAT_1:74
theorem Th75: :: RELAT_1:75
theorem Th76: :: RELAT_1:76
theorem Th77: :: RELAT_1:77
theorem Th78: :: RELAT_1:78
theorem Th79: :: RELAT_1:79
theorem Th80: :: RELAT_1:80
theorem Th81: :: RELAT_1:81
theorem Th82: :: RELAT_1:82
definition
let c
1 be
Relation;
let c
2 be
set ;
func c
1 | c
2 -> Relation means :
Def11:
:: RELAT_1:def 11
for b
1, b
2 being
set holds
(
[b1,b2] in a
3 iff ( b
1 in a
2 &
[b1,b2] in a
1 ) );
existence
ex b1 being Relation st
for b2, b3 being set holds
( [b2,b3] in b1 iff ( b2 in c2 & [b2,b3] in c1 ) )
uniqueness
for b1, b2 being Relation holds
( ( for b3, b4 being set holds
( [b3,b4] in b1 iff ( b3 in c2 & [b3,b4] in c1 ) ) ) & ( for b3, b4 being set holds
( [b3,b4] in b2 iff ( b3 in c2 & [b3,b4] in c1 ) ) ) implies b1 = b2 )
end;
:: deftheorem Def11 defines | RELAT_1:def 11 :
theorem Th83: :: RELAT_1:83
canceled;
theorem Th84: :: RELAT_1:84
canceled;
theorem Th85: :: RELAT_1:85
canceled;
theorem Th86: :: RELAT_1:86
theorem Th87: :: RELAT_1:87
theorem Th88: :: RELAT_1:88
theorem Th89: :: RELAT_1:89
theorem Th90: :: RELAT_1:90
theorem Th91: :: RELAT_1:91
theorem Th92: :: RELAT_1:92
theorem Th93: :: RELAT_1:93
theorem Th94: :: RELAT_1:94
theorem Th95: :: RELAT_1:95
theorem Th96: :: RELAT_1:96
theorem Th97: :: RELAT_1:97
theorem Th98: :: RELAT_1:98
theorem Th99: :: RELAT_1:99
theorem Th100: :: RELAT_1:100
theorem Th101: :: RELAT_1:101
theorem Th102: :: RELAT_1:102
for b
1, b
2 being
set for b
3 being
Relation holds
( b
1 c= b
2 implies
(b3 | b1) | b
2 = b
3 | b
1 )
theorem Th103: :: RELAT_1:103
for b
1, b
2 being
set for b
3 being
Relation holds
( b
1 c= b
2 implies
(b3 | b2) | b
1 = b
3 | b
1 )
theorem Th104: :: RELAT_1:104
for b
1, b
2 being
set for b
3 being
Relation holds
( b
1 c= b
2 implies b
3 | b
1 c= b
3 | b
2 )
theorem Th105: :: RELAT_1:105
for b
1 being
set for b
2, b
3 being
Relation holds
( b
2 c= b
3 implies b
2 | b
1 c= b
3 | b
1 )
theorem Th106: :: RELAT_1:106
for b
1, b
2 being
set for b
3, b
4 being
Relation holds
( b
3 c= b
4 & b
1 c= b
2 implies b
3 | b
1 c= b
4 | b
2 )
theorem Th107: :: RELAT_1:107
theorem Th108: :: RELAT_1:108
theorem Th109: :: RELAT_1:109
for b
1, b
2 being
set for b
3 being
Relation holds b
3 | (b1 \ b2) = (b3 | b1) \ (b3 | b2)
theorem Th110: :: RELAT_1:110
theorem Th111: :: RELAT_1:111
theorem Th112: :: RELAT_1:112
for b
1 being
set for b
2, b
3 being
Relation holds
(b2 * b3) | b
1 = (b2 | b1) * b
3
definition
let c
1 be
set ;
let c
2 be
Relation;
func c
1 | c
2 -> Relation means :
Def12:
:: RELAT_1:def 12
for b
1, b
2 being
set holds
(
[b1,b2] in a
3 iff ( b
2 in a
1 &
[b1,b2] in a
2 ) );
existence
ex b1 being Relation st
for b2, b3 being set holds
( [b2,b3] in b1 iff ( b3 in c1 & [b2,b3] in c2 ) )
uniqueness
for b1, b2 being Relation holds
( ( for b3, b4 being set holds
( [b3,b4] in b1 iff ( b4 in c1 & [b3,b4] in c2 ) ) ) & ( for b3, b4 being set holds
( [b3,b4] in b2 iff ( b4 in c1 & [b3,b4] in c2 ) ) ) implies b1 = b2 )
end;
:: deftheorem Def12 defines | RELAT_1:def 12 :
for b
1 being
set for b
2, b
3 being
Relation holds
( b
3 = b
1 | b
2 iff for b
4, b
5 being
set holds
(
[b4,b5] in b
3 iff ( b
5 in b
1 &
[b4,b5] in b
2 ) ) );
theorem Th113: :: RELAT_1:113
canceled;
theorem Th114: :: RELAT_1:114
canceled;
theorem Th115: :: RELAT_1:115
theorem Th116: :: RELAT_1:116
theorem Th117: :: RELAT_1:117
theorem Th118: :: RELAT_1:118
theorem Th119: :: RELAT_1:119
theorem Th120: :: RELAT_1:120
theorem Th121: :: RELAT_1:121
theorem Th122: :: RELAT_1:122
theorem Th123: :: RELAT_1:123
theorem Th124: :: RELAT_1:124
theorem Th125: :: RELAT_1:125
theorem Th126: :: RELAT_1:126
theorem Th127: :: RELAT_1:127
theorem Th128: :: RELAT_1:128
theorem Th129: :: RELAT_1:129
for b
1, b
2 being
set for b
3 being
Relation holds
( b
1 c= b
2 implies b
2 | (b1 | b3) = b
1 | b
3 )
theorem Th130: :: RELAT_1:130
for b
1, b
2 being
set for b
3 being
Relation holds
( b
1 c= b
2 implies b
1 | (b2 | b3) = b
1 | b
3 )
theorem Th131: :: RELAT_1:131
for b
1, b
2 being
set for b
3 being
Relation holds
( b
1 c= b
2 implies b
1 | b
3 c= b
2 | b
3 )
theorem Th132: :: RELAT_1:132
for b
1 being
set for b
2, b
3 being
Relation holds
( b
2 c= b
3 implies b
1 | b
2 c= b
1 | b
3 )
theorem Th133: :: RELAT_1:133
for b
1, b
2 being
set for b
3, b
4 being
Relation holds
( b
3 c= b
4 & b
1 c= b
2 implies b
1 | b
3 c= b
2 | b
4 )
theorem Th134: :: RELAT_1:134
theorem Th135: :: RELAT_1:135
theorem Th136: :: RELAT_1:136
for b
1, b
2 being
set for b
3 being
Relation holds
(b1 \ b2) | b
3 = (b1 | b3) \ (b2 | b3)
theorem Th137: :: RELAT_1:137
theorem Th138: :: RELAT_1:138
theorem Th139: :: RELAT_1:139
for b
1 being
set for b
2, b
3 being
Relation holds b
1 | (b2 * b3) = b
2 * (b1 | b3)
theorem Th140: :: RELAT_1:140
for b
1, b
2 being
set for b
3 being
Relation holds
(b1 | b3) | b
2 = b
1 | (b3 | b2)
:: deftheorem Def13 defines .: RELAT_1:def 13 :
for b
1 being
Relationfor b
2, b
3 being
set holds
( b
3 = b
1 .: b
2 iff for b
4 being
set holds
( b
4 in b
3 iff ex b
5 being
set st
(
[b5,b4] in b
1 & b
5 in b
2 ) ) );
theorem Th141: :: RELAT_1:141
canceled;
theorem Th142: :: RELAT_1:142
canceled;
theorem Th143: :: RELAT_1:143
theorem Th144: :: RELAT_1:144
theorem Th145: :: RELAT_1:145
theorem Th146: :: RELAT_1:146
theorem Th147: :: RELAT_1:147
theorem Th148: :: RELAT_1:148
theorem Th149: :: RELAT_1:149
theorem Th150: :: RELAT_1:150
theorem Th151: :: RELAT_1:151
theorem Th152: :: RELAT_1:152
theorem Th153: :: RELAT_1:153
theorem Th154: :: RELAT_1:154
theorem Th155: :: RELAT_1:155
theorem Th156: :: RELAT_1:156
theorem Th157: :: RELAT_1:157
theorem Th158: :: RELAT_1:158
for b
1, b
2 being
set for b
3, b
4 being
Relation holds
( b
3 c= b
4 & b
1 c= b
2 implies b
3 .: b
1 c= b
4 .: b
2 )
theorem Th159: :: RELAT_1:159
theorem Th160: :: RELAT_1:160
theorem Th161: :: RELAT_1:161
theorem Th162: :: RELAT_1:162
for b
1 being
Relationfor b
2, b
3 being
set holds
( b
2 c= b
3 implies
(b1 | b3) .: b
2 = b
1 .: b
2 )
theorem Th163: :: RELAT_1:163
:: deftheorem Def14 defines " RELAT_1:def 14 :
for b
1 being
Relationfor b
2, b
3 being
set holds
( b
3 = b
1 " b
2 iff for b
4 being
set holds
( b
4 in b
3 iff ex b
5 being
set st
(
[b4,b5] in b
1 & b
5 in b
2 ) ) );
theorem Th164: :: RELAT_1:164
canceled;
theorem Th165: :: RELAT_1:165
canceled;
theorem Th166: :: RELAT_1:166
theorem Th167: :: RELAT_1:167
theorem Th168: :: RELAT_1:168
theorem Th169: :: RELAT_1:169
theorem Th170: :: RELAT_1:170
theorem Th171: :: RELAT_1:171
theorem Th172: :: RELAT_1:172
theorem Th173: :: RELAT_1:173
theorem Th174: :: RELAT_1:174
theorem Th175: :: RELAT_1:175
theorem Th176: :: RELAT_1:176
theorem Th177: :: RELAT_1:177
theorem Th178: :: RELAT_1:178
for b
1, b
2 being
set for b
3 being
Relation holds
( b
1 c= b
2 implies b
3 " b
1 c= b
3 " b
2 )
theorem Th179: :: RELAT_1:179
for b
1 being
set for b
2, b
3 being
Relation holds
( b
2 c= b
3 implies b
2 " b
1 c= b
3 " b
1 )
theorem Th180: :: RELAT_1:180
for b
1, b
2 being
set for b
3, b
4 being
Relation holds
( b
3 c= b
4 & b
1 c= b
2 implies b
3 " b
1 c= b
4 " b
2 )
theorem Th181: :: RELAT_1:181
for b
1 being
set for b
2, b
3 being
Relation holds
(b2 * b3) " b
1 = b
2 " (b3 " b1)
theorem Th182: :: RELAT_1:182
theorem Th183: :: RELAT_1:183
:: deftheorem Def15 defines empty-yielding RELAT_1:def 15 :
theorem Th184: :: RELAT_1:184
theorem Th185: :: RELAT_1:185
for b
1, b
2 being
Relationfor b
3, b
4 being
set holds
( b
1 | b
3 = b
2 | b
3 & b
1 | b
4 = b
2 | b
4 implies b
1 | (b3 \/ b4) = b
2 | (b3 \/ b4) )
theorem Th186: :: RELAT_1:186
theorem Th187: :: RELAT_1:187
theorem Th188: :: RELAT_1:188
for b
1, b
2 being
Relationfor b
3, b
4 being
set holds
( b
3 c= b
4 & b
1 | b
4 = b
2 | b
4 implies b
1 | b
3 = b
2 | b
3 )
theorem Th189: :: RELAT_1:189
theorem Th190: :: RELAT_1:190