:: RELSET_1 semantic presentation
:: deftheorem Def1 defines Relation RELSET_1:def 1 :
theorem Th1: :: RELSET_1:1
canceled;
theorem Th2: :: RELSET_1:2
canceled;
theorem Th3: :: RELSET_1:3
canceled;
theorem Th4: :: RELSET_1:4
theorem Th5: :: RELSET_1:5
canceled;
theorem Th6: :: RELSET_1:6
for
a,
X,
Y being
set for
R being
Relation of
X,
Y st
a in R holds
ex
x,
y being
set st
(
a = [x,y] &
x in X &
y in Y )
theorem Th7: :: RELSET_1:7
canceled;
theorem Th8: :: RELSET_1:8
theorem Th9: :: RELSET_1:9
theorem Th10: :: RELSET_1:10
theorem Th11: :: RELSET_1:11
theorem Th12: :: RELSET_1:12
theorem Th13: :: RELSET_1:13
theorem Th14: :: RELSET_1:14
theorem Th15: :: RELSET_1:15
theorem Th16: :: RELSET_1:16
theorem Th17: :: RELSET_1:17
theorem Th18: :: RELSET_1:18
canceled;
theorem Th19: :: RELSET_1:19
theorem Th20: :: RELSET_1:20
canceled;
theorem Th21: :: RELSET_1:21
canceled;
theorem Th22: :: RELSET_1:22
for
Y,
X being
set for
R being
Relation of
X,
Y holds
( ( for
x being
set st
x in X holds
ex
y being
set st
[x,y] in R ) iff
dom R = X )
theorem Th23: :: RELSET_1:23
for
X,
Y being
set for
R being
Relation of
X,
Y holds
( ( for
y being
set st
y in Y holds
ex
x being
set st
[x,y] in R ) iff
rng R = Y )
theorem Th24: :: RELSET_1:24
theorem Th25: :: RELSET_1:25
theorem Th26: :: RELSET_1:26
theorem Th27: :: RELSET_1:27
theorem Th28: :: RELSET_1:28
theorem Th29: :: RELSET_1:29
theorem Th30: :: RELSET_1:30
theorem Th31: :: RELSET_1:31
theorem Th32: :: RELSET_1:32
theorem Th33: :: RELSET_1:33
theorem Th34: :: RELSET_1:34
for
Y,
X,
X1 being
set for
R being
Relation of
X,
Y st
X c= X1 holds
R | X1 = R
theorem Th35: :: RELSET_1:35
theorem Th36: :: RELSET_1:36
for
X,
Y,
Y1 being
set for
R being
Relation of
X,
Y st
Y c= Y1 holds
Y1 | R = R
theorem Th37: :: RELSET_1:37
canceled;
theorem Th38: :: RELSET_1:38
theorem Th39: :: RELSET_1:39
theorem Th40: :: RELSET_1:40
canceled;
theorem Th41: :: RELSET_1:41
canceled;
theorem Th42: :: RELSET_1:42
canceled;
theorem Th43: :: RELSET_1:43
canceled;
theorem Th44: :: RELSET_1:44
canceled;
theorem Th45: :: RELSET_1:45
canceled;
theorem Th46: :: RELSET_1:46
theorem Th47: :: RELSET_1:47
theorem Th48: :: RELSET_1:48
theorem Th49: :: RELSET_1:49
theorem Th50: :: RELSET_1:50
theorem Th51: :: RELSET_1:51
theorem Th52: :: RELSET_1:52
theorem Th53: :: RELSET_1:53
theorem Th54: :: RELSET_1:54