:: FUNCT_1 semantic presentation
:: deftheorem Def1 defines Function-like FUNCT_1:def 1 :
theorem Th1: :: FUNCT_1:1
canceled;
theorem Th2: :: FUNCT_1:2
:: deftheorem Def2 FUNCT_1:def 2 :
canceled;
:: deftheorem Def3 FUNCT_1:def 3 :
canceled;
:: deftheorem Def4 defines . FUNCT_1:def 4 :
for b
1 being
Functionfor b
2, b
3 being
set holds
( ( b
2 in dom b
1 implies ( b
3 = b
1 . b
2 iff
[b2,b3] in b
1 ) ) & ( not b
2 in dom b
1 implies ( b
3 = b
1 . b
2 iff b
3 = {} ) ) );
theorem Th3: :: FUNCT_1:3
canceled;
theorem Th4: :: FUNCT_1:4
canceled;
theorem Th5: :: FUNCT_1:5
canceled;
theorem Th6: :: FUNCT_1:6
canceled;
theorem Th7: :: FUNCT_1:7
canceled;
theorem Th8: :: FUNCT_1:8
theorem Th9: :: FUNCT_1:9
for b
1, b
2 being
Function holds
(
dom b
1 = dom b
2 & ( for b
3 being
set holds
( b
3 in dom b
1 implies b
1 . b
3 = b
2 . b
3 ) ) implies b
1 = b
2 )
:: deftheorem Def5 defines rng FUNCT_1:def 5 :
for b
1 being
Functionfor 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
( b
4 in dom b
1 & b
3 = b
1 . b
4 ) ) );
theorem Th10: :: FUNCT_1:10
canceled;
theorem Th11: :: FUNCT_1:11
canceled;
theorem Th12: :: FUNCT_1:12
theorem Th13: :: FUNCT_1:13
canceled;
theorem Th14: :: FUNCT_1:14
scheme :: FUNCT_1:sch 2
s2{ F
1()
-> set , P
1[
set ,
set ] } :
ex b
1 being
Function st
(
dom b
1 = F
1() & ( for b
2 being
set holds
( b
2 in F
1() implies P
1[b
2,b
1 . b
2] ) ) )
provided
E7:
for b
1, b
2, b
3 being
set holds
( b
1 in F
1() & P
1[b
1,b
2] & P
1[b
1,b
3] implies b
2 = b
3 )
and
E8:
for b
1 being
set holds
not ( b
1 in F
1() & ( for b
2 being
set holds
not P
1[b
1,b
2] ) )
theorem Th15: :: FUNCT_1:15
theorem Th16: :: FUNCT_1:16
for b
1 being
set holds
( ( for b
2, b
3 being
Function holds
(
dom b
2 = b
1 &
dom b
3 = b
1 implies b
2 = b
3 ) ) implies b
1 = {} )
theorem Th17: :: FUNCT_1:17
theorem Th18: :: FUNCT_1:18
for b
1, b
2 being
set holds
not ( not ( not b
1 <> {} & not b
2 = {} ) & ( for b
3 being
Function holds
not ( b
2 = dom b
3 &
rng b
3 c= b
1 ) ) )
theorem Th19: :: FUNCT_1:19
for b
1 being
set for b
2 being
Function holds
( ( for b
3 being
set holds
not ( b
3 in b
1 & ( for b
4 being
set holds
not ( b
4 in dom b
2 & b
3 = b
2 . b
4 ) ) ) ) implies b
1 c= rng b
2 )
theorem Th20: :: FUNCT_1:20
for b
1, b
2, b
3 being
Function holds
( ( for b
4 being
set holds
( b
4 in dom b
3 iff ( b
4 in dom b
1 & b
1 . b
4 in dom b
2 ) ) ) & ( for b
4 being
set holds
( b
4 in dom b
3 implies b
3 . b
4 = b
2 . (b1 . b4) ) ) implies b
3 = b
2 * b
1 )
theorem Th21: :: FUNCT_1:21
theorem Th22: :: FUNCT_1:22
for b
1 being
set for b
2, b
3 being
Function holds
( b
1 in dom (b2 * b3) implies
(b2 * b3) . b
1 = b
2 . (b3 . b1) )
theorem Th23: :: FUNCT_1:23
for b
1 being
set for b
2, b
3 being
Function holds
( b
1 in dom b
2 implies
(b3 * b2) . b
1 = b
3 . (b2 . b1) )
theorem Th24: :: FUNCT_1:24
canceled;
theorem Th25: :: FUNCT_1:25
theorem Th26: :: FUNCT_1:26
canceled;
theorem Th27: :: FUNCT_1:27
theorem Th28: :: FUNCT_1:28
canceled;
theorem Th29: :: FUNCT_1:29
canceled;
theorem Th30: :: FUNCT_1:30
canceled;
theorem Th31: :: FUNCT_1:31
canceled;
theorem Th32: :: FUNCT_1:32
canceled;
theorem Th33: :: FUNCT_1:33
theorem Th34: :: FUNCT_1:34
for b
1 being
set for b
2 being
Function holds
( b
2 = id b
1 iff (
dom b
2 = b
1 & ( for b
3 being
set holds
( b
3 in b
1 implies b
2 . b
3 = b
3 ) ) ) )
theorem Th35: :: FUNCT_1:35
for b
1, b
2 being
set holds
( b
2 in b
1 implies
(id b1) . b
2 = b
2 )
by Th34;
theorem Th36: :: FUNCT_1:36
canceled;
theorem Th37: :: FUNCT_1:37
theorem Th38: :: FUNCT_1:38
theorem Th39: :: FUNCT_1:39
canceled;
theorem Th40: :: FUNCT_1:40
theorem Th41: :: FUNCT_1:41
canceled;
theorem Th42: :: FUNCT_1:42
canceled;
theorem Th43: :: FUNCT_1:43
theorem Th44: :: FUNCT_1:44
:: deftheorem Def6 FUNCT_1:def 6 :
canceled;
:: deftheorem Def7 FUNCT_1:def 7 :
canceled;
:: deftheorem Def8 defines one-to-one FUNCT_1:def 8 :
theorem Th45: :: FUNCT_1:45
canceled;
theorem Th46: :: FUNCT_1:46
theorem Th47: :: FUNCT_1:47
theorem Th48: :: FUNCT_1:48
theorem Th49: :: FUNCT_1:49
theorem Th50: :: FUNCT_1:50
theorem Th51: :: FUNCT_1:51
theorem Th52: :: FUNCT_1:52
theorem Th53: :: FUNCT_1:53
:: deftheorem Def9 defines " FUNCT_1:def 9 :
theorem Th54: :: FUNCT_1:54
theorem Th55: :: FUNCT_1:55
theorem Th56: :: FUNCT_1:56
theorem Th57: :: FUNCT_1:57
theorem Th58: :: FUNCT_1:58
theorem Th59: :: FUNCT_1:59
theorem Th60: :: FUNCT_1:60
theorem Th61: :: FUNCT_1:61
theorem Th62: :: FUNCT_1:62
Lemma27:
for b1 being set
for b2, b3, b4 being Function holds
( rng b2 = b1 & b3 * b2 = id (dom b4) & b4 * b3 = id b1 implies b4 = b2 )
theorem Th63: :: FUNCT_1:63
theorem Th64: :: FUNCT_1:64
theorem Th65: :: FUNCT_1:65
theorem Th66: :: FUNCT_1:66
theorem Th67: :: FUNCT_1:67
theorem Th68: :: FUNCT_1:68
for b
1 being
set for b
2, b
3 being
Function holds
( b
2 = b
3 | b
1 iff (
dom b
2 = (dom b3) /\ b
1 & ( for b
4 being
set holds
( b
4 in dom b
2 implies b
2 . b
4 = b
3 . b
4 ) ) ) )
theorem Th69: :: FUNCT_1:69
canceled;
theorem Th70: :: FUNCT_1:70
theorem Th71: :: FUNCT_1:71
Lemma30:
for b1, b2 being set
for b3 being Function holds
( b2 in dom (b3 | b1) iff ( b2 in dom b3 & b2 in b1 ) )
theorem Th72: :: FUNCT_1:72
for b
1, b
2 being
set for b
3 being
Function holds
( b
2 in b
1 implies
(b3 | b1) . b
2 = b
3 . b
2 )
theorem Th73: :: FUNCT_1:73
theorem Th74: :: FUNCT_1:74
canceled;
theorem Th75: :: FUNCT_1:75
canceled;
theorem Th76: :: FUNCT_1:76
canceled;
theorem Th77: :: FUNCT_1:77
canceled;
theorem Th78: :: FUNCT_1:78
canceled;
theorem Th79: :: FUNCT_1:79
canceled;
theorem Th80: :: FUNCT_1:80
canceled;
theorem Th81: :: FUNCT_1:81
canceled;
theorem Th82: :: FUNCT_1:82
theorem Th83: :: FUNCT_1:83
canceled;
theorem Th84: :: FUNCT_1:84
theorem Th85: :: FUNCT_1:85
for b
1 being
set for b
2, b
3 being
Function holds
( b
2 = b
1 | b
3 iff ( ( for b
4 being
set holds
( b
4 in dom b
2 iff ( b
4 in dom b
3 & b
3 . b
4 in b
1 ) ) ) & ( for b
4 being
set holds
( b
4 in dom b
2 implies b
2 . b
4 = b
3 . b
4 ) ) ) )
theorem Th86: :: FUNCT_1:86
theorem Th87: :: FUNCT_1:87
theorem Th88: :: FUNCT_1:88
canceled;
theorem Th89: :: FUNCT_1:89
theorem Th90: :: FUNCT_1:90
canceled;
theorem Th91: :: FUNCT_1:91
canceled;
theorem Th92: :: FUNCT_1:92
canceled;
theorem Th93: :: FUNCT_1:93
canceled;
theorem Th94: :: FUNCT_1:94
canceled;
theorem Th95: :: FUNCT_1:95
canceled;
theorem Th96: :: FUNCT_1:96
canceled;
theorem Th97: :: FUNCT_1:97
theorem Th98: :: FUNCT_1:98
canceled;
theorem Th99: :: FUNCT_1:99
:: deftheorem Def10 FUNCT_1:def 10 :
canceled;
:: deftheorem Def11 FUNCT_1:def 11 :
canceled;
:: deftheorem Def12 defines .: FUNCT_1:def 12 :
for b
1 being
Functionfor b
2 being
set for 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
( b
5 in dom b
1 & b
5 in b
2 & b
4 = b
1 . b
5 ) ) );
theorem Th100: :: FUNCT_1:100
canceled;
theorem Th101: :: FUNCT_1:101
canceled;
theorem Th102: :: FUNCT_1:102
canceled;
theorem Th103: :: FUNCT_1:103
canceled;
theorem Th104: :: FUNCT_1:104
canceled;
theorem Th105: :: FUNCT_1:105
canceled;
theorem Th106: :: FUNCT_1:106
canceled;
theorem Th107: :: FUNCT_1:107
canceled;
theorem Th108: :: FUNCT_1:108
canceled;
theorem Th109: :: FUNCT_1:109
canceled;
theorem Th110: :: FUNCT_1:110
canceled;
theorem Th111: :: FUNCT_1:111
canceled;
theorem Th112: :: FUNCT_1:112
canceled;
theorem Th113: :: FUNCT_1:113
canceled;
theorem Th114: :: FUNCT_1:114
canceled;
theorem Th115: :: FUNCT_1:115
canceled;
theorem Th116: :: FUNCT_1:116
canceled;
theorem Th117: :: FUNCT_1:117
theorem Th118: :: FUNCT_1:118
theorem Th119: :: FUNCT_1:119
canceled;
theorem Th120: :: FUNCT_1:120
theorem Th121: :: FUNCT_1:121
theorem Th122: :: FUNCT_1:122
theorem Th123: :: FUNCT_1:123
theorem Th124: :: FUNCT_1:124
theorem Th125: :: FUNCT_1:125
theorem Th126: :: FUNCT_1:126
:: deftheorem Def13 defines " FUNCT_1:def 13 :
for b
1 being
Functionfor b
2 being
set for b
3 being
set holds
( b
3 = b
1 " b
2 iff for b
4 being
set holds
( b
4 in b
3 iff ( b
4 in dom b
1 & b
1 . b
4 in b
2 ) ) );
theorem Th127: :: FUNCT_1:127
canceled;
theorem Th128: :: FUNCT_1:128
canceled;
theorem Th129: :: FUNCT_1:129
canceled;
theorem Th130: :: FUNCT_1:130
canceled;
theorem Th131: :: FUNCT_1:131
canceled;
theorem Th132: :: FUNCT_1:132
canceled;
theorem Th133: :: FUNCT_1:133
canceled;
theorem Th134: :: FUNCT_1:134
canceled;
theorem Th135: :: FUNCT_1:135
canceled;
theorem Th136: :: FUNCT_1:136
canceled;
theorem Th137: :: FUNCT_1:137
theorem Th138: :: FUNCT_1:138
for b
1, b
2 being
set for b
3 being
Function holds b
3 " (b1 \ b2) = (b3 " b1) \ (b3 " b2)
theorem Th139: :: FUNCT_1:139
theorem Th140: :: FUNCT_1:140
canceled;
theorem Th141: :: FUNCT_1:141
theorem Th142: :: FUNCT_1:142
theorem Th143: :: FUNCT_1:143
theorem Th144: :: FUNCT_1:144
theorem Th145: :: FUNCT_1:145
theorem Th146: :: FUNCT_1:146
theorem Th147: :: FUNCT_1:147
theorem Th148: :: FUNCT_1:148
theorem Th149: :: FUNCT_1:149
theorem Th150: :: FUNCT_1:150
theorem Th151: :: FUNCT_1:151
theorem Th152: :: FUNCT_1:152
theorem Th153: :: FUNCT_1:153
theorem Th154: :: FUNCT_1:154
theorem Th155: :: FUNCT_1:155
theorem Th156: :: FUNCT_1:156
theorem Th157: :: FUNCT_1:157
theorem Th158: :: FUNCT_1:158
theorem Th159: :: FUNCT_1:159
theorem Th160: :: FUNCT_1:160
theorem Th161: :: FUNCT_1:161
theorem Th162: :: FUNCT_1:162
:: deftheorem Def14 defines empty-yielding FUNCT_1:def 14 :
:: deftheorem Def15 defines non-empty FUNCT_1:def 15 :
:: deftheorem Def16 defines constant FUNCT_1:def 16 :
theorem Th163: :: FUNCT_1:163
theorem Th164: :: FUNCT_1:164