:: FUNCT_6 semantic presentation
theorem Th1: :: FUNCT_6:1
theorem Th2: :: FUNCT_6:2
theorem Th3: :: FUNCT_6:3
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
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
X,
Y,
Z,
V1,
V2 being
set for
f being
Function st (
curry f in Funcs X,
(Funcs Y,Z) or
curry' f in Funcs Y,
(Funcs X,Z) ) &
dom f c= [:V1,V2:] holds
f in Funcs [:X,Y:],
Z
theorem Th22: :: FUNCT_6:22
for
X,
Y,
Z,
V1,
V2 being
set for
f being
Function st (
uncurry f in Funcs [:X,Y:],
Z or
uncurry' f in Funcs [:Y,X:],
Z ) &
rng f c= PFuncs V1,
V2 &
dom f = X holds
f in Funcs X,
(Funcs Y,Z)
theorem Th23: :: FUNCT_6:23
theorem Th24: :: FUNCT_6:24
theorem Th25: :: FUNCT_6:25
for
X,
Y,
Z,
V1,
V2 being
set for
f being
Function st (
curry f in PFuncs X,
(PFuncs Y,Z) or
curry' f in PFuncs Y,
(PFuncs X,Z) ) &
dom f c= [:V1,V2:] holds
f in PFuncs [:X,Y:],
Z
theorem Th26: :: FUNCT_6:26
for
X,
Y,
Z,
V1,
V2 being
set for
f being
Function st (
uncurry f in PFuncs [:X,Y:],
Z or
uncurry' f in PFuncs [:Y,X:],
Z ) &
rng f c= PFuncs V1,
V2 &
dom f c= X holds
f in PFuncs X,
(PFuncs Y,Z)
:: deftheorem Def1 defines SubFuncs FUNCT_6:def 1 :
theorem Th27: :: FUNCT_6:27
theorem Th28: :: FUNCT_6:28
Lemma72:
for X being set st ( for x being set st x in X holds
x is Function ) holds
SubFuncs X = X
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
f,
g,
h being
Function holds
(
doms <*f,g,h*> = <*(dom f),(dom g),(dom h)*> &
rngs <*f,g,h*> = <*(rng f),(rng g),(rng h)*> )
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
Lemma101:
for f being Function holds rng (Frege f) c= product (rngs f)
theorem Th57: :: FUNCT_6:57
Lemma104:
for f being Function holds product (rngs f) c= rng (Frege f)
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
Lemma128:
for x, y, z being set
for f, g being Function st [x,y] in dom f & g = (curry f) . x & z in dom g holds
[x,z] in dom f
theorem Th77: :: FUNCT_6:77
:: deftheorem Def9 defines Funcs FUNCT_6:def 9 :
Lemma166:
for X being set
for f being Function st f <> {} & X <> {} holds
product (Funcs X,f), Funcs X,(product f) 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
Lemma181:
for f being Function st dom f = {} holds
commute f = {}
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