:: FUNCT_6 semantic presentation
theorem Th1: :: FUNCT_6:1
theorem Th2: :: FUNCT_6:2
theorem Th3: :: FUNCT_6:3
for b
1, b
2, b
3, b
4 being
set holds
( b
1 in product <*b2,b3,b4*> iff ex b
5, b
6, b
7 being
set st
( b
5 in b
2 & b
6 in b
3 & b
7 in b
4 & b
1 = <*b5,b6,b7*> ) )
theorem Th4: :: FUNCT_6:4
theorem Th5: :: FUNCT_6:5
theorem Th6: :: FUNCT_6:6
theorem Th7: :: FUNCT_6:7
theorem Th8: :: FUNCT_6:8
theorem Th9: :: FUNCT_6:9
theorem Th10: :: FUNCT_6:10
theorem Th11: :: FUNCT_6:11
for b
1 being
set for b
2 being
Function holds
not ( b
1 in dom (~ b2) & ( for b
3, b
4 being
set holds
not b
1 = [b3,b4] ) )
theorem Th12: :: FUNCT_6:12
theorem Th13: :: FUNCT_6:13
theorem Th14: :: FUNCT_6:14
theorem Th15: :: FUNCT_6:15
theorem Th16: :: FUNCT_6:16
theorem Th17: :: FUNCT_6:17
theorem Th18: :: FUNCT_6:18
theorem Th19: :: FUNCT_6:19
theorem Th20: :: FUNCT_6:20
theorem Th21: :: FUNCT_6:21
for b
1, b
2, b
3, b
4, b
5 being
set for b
6 being
Function holds
( (
curry b
6 in Funcs b
1,
(Funcs b2,b3) or
curry' b
6 in Funcs b
2,
(Funcs b1,b3) ) &
dom b
6 c= [:b4,b5:] implies b
6 in Funcs [:b1,b2:],b
3 )
theorem Th22: :: FUNCT_6:22
for b
1, b
2, b
3, b
4, b
5 being
set for b
6 being
Function holds
( (
uncurry b
6 in Funcs [:b1,b2:],b
3 or
uncurry' b
6 in Funcs [:b2,b1:],b
3 ) &
rng b
6 c= PFuncs b
4,b
5 &
dom b
6 = b
1 implies b
6 in Funcs b
1,
(Funcs b2,b3) )
theorem Th23: :: FUNCT_6:23
theorem Th24: :: FUNCT_6:24
theorem Th25: :: FUNCT_6:25
for b
1, b
2, b
3, b
4, b
5 being
set for b
6 being
Function holds
( (
curry b
6 in PFuncs b
1,
(PFuncs b2,b3) or
curry' b
6 in PFuncs b
2,
(PFuncs b1,b3) ) &
dom b
6 c= [:b4,b5:] implies b
6 in PFuncs [:b1,b2:],b
3 )
theorem Th26: :: FUNCT_6:26
for b
1, b
2, b
3, b
4, b
5 being
set for b
6 being
Function holds
( (
uncurry b
6 in PFuncs [:b1,b2:],b
3 or
uncurry' b
6 in PFuncs [:b2,b1:],b
3 ) &
rng b
6 c= PFuncs b
4,b
5 &
dom b
6 c= b
1 implies b
6 in PFuncs b
1,
(PFuncs b2,b3) )
:: deftheorem Def1 defines SubFuncs FUNCT_6:def 1 :
theorem Th27: :: FUNCT_6:27
theorem Th28: :: FUNCT_6:28
Lemma18:
for b1 being set holds
( ( for b2 being set holds
( b2 in b1 implies b2 is Function ) ) implies SubFuncs b1 = b1 )
theorem Th29: :: FUNCT_6:29
theorem Th30: :: FUNCT_6:30
:: deftheorem Def2 defines doms FUNCT_6:def 2 :
:: deftheorem Def3 defines rngs FUNCT_6:def 3 :
:: deftheorem Def4 defines meet FUNCT_6:def 4 :
theorem Th31: :: FUNCT_6:31
theorem Th32: :: FUNCT_6:32
theorem Th33: :: FUNCT_6:33
theorem Th34: :: FUNCT_6:34
theorem Th35: :: FUNCT_6:35
for b
1, b
2, b
3 being
Function holds
(
doms <*b1,b2,b3*> = <*(dom b1),(dom b2),(dom b3)*> &
rngs <*b1,b2,b3*> = <*(rng b1),(rng b2),(rng b3)*> )
theorem Th36: :: FUNCT_6:36
theorem Th37: :: FUNCT_6:37
theorem Th38: :: FUNCT_6:38
theorem Th39: :: FUNCT_6:39
theorem Th40: :: FUNCT_6:40
theorem Th41: :: FUNCT_6:41
theorem Th42: :: FUNCT_6:42
theorem Th43: :: FUNCT_6:43
:: deftheorem Def5 defines .. FUNCT_6:def 5 :
theorem Th44: :: FUNCT_6:44
theorem Th45: :: FUNCT_6:45
theorem Th46: :: FUNCT_6:46
theorem Th47: :: FUNCT_6:47
theorem Th48: :: FUNCT_6:48
:: deftheorem Def6 defines <: FUNCT_6:def 6 :
theorem Th49: :: FUNCT_6:49
theorem Th50: :: FUNCT_6:50
theorem Th51: :: FUNCT_6:51
theorem Th52: :: FUNCT_6:52
theorem Th53: :: FUNCT_6:53
theorem Th54: :: FUNCT_6:54
theorem Th55: :: FUNCT_6:55
:: deftheorem Def7 defines Frege FUNCT_6:def 7 :
theorem Th56: :: FUNCT_6:56
Lemma37:
for b1 being Function holds rng (Frege b1) c= product (rngs b1)
theorem Th57: :: FUNCT_6:57
Lemma39:
for b1 being Function holds product (rngs b1) c= rng (Frege b1)
theorem Th58: :: FUNCT_6:58
theorem Th59: :: FUNCT_6:59
theorem Th60: :: FUNCT_6:60
theorem Th61: :: FUNCT_6:61
theorem Th62: :: FUNCT_6:62
theorem Th63: :: FUNCT_6:63
theorem Th64: :: FUNCT_6:64
theorem Th65: :: FUNCT_6:65
theorem Th66: :: FUNCT_6:66
theorem Th67: :: FUNCT_6:67
theorem Th68: :: FUNCT_6:68
theorem Th69: :: FUNCT_6:69
theorem Th70: :: FUNCT_6:70
theorem Th71: :: FUNCT_6:71
:: deftheorem Def8 defines Funcs FUNCT_6:def 8 :
theorem Th72: :: FUNCT_6:72
theorem Th73: :: FUNCT_6:73
theorem Th74: :: FUNCT_6:74
theorem Th75: :: FUNCT_6:75
theorem Th76: :: FUNCT_6:76
Lemma47:
for b1, b2, b3 being set
for b4, b5 being Function holds
( [b1,b2] in dom b4 & b5 = (curry b4) . b1 & b3 in dom b5 implies [b1,b3] in dom b4 )
theorem Th77: :: FUNCT_6:77
:: deftheorem Def9 defines Funcs FUNCT_6:def 9 :
Lemma49:
for b1 being set
for b2 being Function holds
( b2 <> {} & b1 <> {} implies product (Funcs b1,b2), Funcs b1,(product b2) are_equipotent )
theorem Th78: :: FUNCT_6:78
theorem Th79: :: FUNCT_6:79
theorem Th80: :: FUNCT_6:80
theorem Th81: :: FUNCT_6:81
theorem Th82: :: FUNCT_6:82
theorem Th83: :: FUNCT_6:83
:: deftheorem Def10 FUNCT_6:def 10 :
canceled;
:: deftheorem Def11 FUNCT_6:def 11 :
canceled;
:: deftheorem Def12 defines commute FUNCT_6:def 12 :
theorem Th84: :: FUNCT_6:84
theorem Th85: :: FUNCT_6:85
theorem Th86: :: FUNCT_6:86
theorem Th87: :: FUNCT_6:87
Lemma53:
for b1 being Function holds
( dom b1 = {} implies commute b1 = {} )
by RELAT_1:64, FUNCT_5:49, FUNCT_5:50;
theorem Th88: :: FUNCT_6:88
theorem Th89: :: FUNCT_6:89
theorem Th90: :: FUNCT_6:90