:: PARTFUN1 semantic presentation
theorem Th1: :: PARTFUN1:1
theorem Th2: :: PARTFUN1:2
theorem Th3: :: PARTFUN1:3
Lemma46:
dom {} = {}
;
theorem Th4: :: PARTFUN1:4
canceled;
theorem Th5: :: PARTFUN1:5
canceled;
theorem Th6: :: PARTFUN1:6
canceled;
theorem Th7: :: PARTFUN1:7
canceled;
theorem Th8: :: PARTFUN1:8
canceled;
theorem Th9: :: PARTFUN1:9
canceled;
theorem Th10: :: PARTFUN1:10
canceled;
Lemma49:
for X, Y being set
for R being Relation holds
( R is Relation of X,Y iff ( dom R c= X & rng R c= Y ) )
by RELSET_1:11, RELSET_1:12;
theorem Th11: :: PARTFUN1:11
canceled;
theorem Th12: :: PARTFUN1:12
canceled;
theorem Th13: :: PARTFUN1:13
canceled;
theorem Th14: :: PARTFUN1:14
canceled;
theorem Th15: :: PARTFUN1:15
canceled;
theorem Th16: :: PARTFUN1:16
canceled;
theorem Th17: :: PARTFUN1:17
canceled;
theorem Th18: :: PARTFUN1:18
canceled;
theorem Th19: :: PARTFUN1:19
canceled;
theorem Th20: :: PARTFUN1:20
canceled;
theorem Th21: :: PARTFUN1:21
canceled;
theorem Th22: :: PARTFUN1:22
canceled;
theorem Th23: :: PARTFUN1:23
canceled;
theorem Th24: :: PARTFUN1:24
theorem Th25: :: PARTFUN1:25
theorem Th26: :: PARTFUN1:26
theorem Th27: :: PARTFUN1:27
theorem Th28: :: PARTFUN1:28
theorem Th29: :: PARTFUN1:29
theorem Th30: :: PARTFUN1:30
theorem Th31: :: PARTFUN1:31
theorem Th32: :: PARTFUN1:32
theorem Th33: :: PARTFUN1:33
theorem Th34: :: PARTFUN1:34
theorem Th35: :: PARTFUN1:35
scheme :: PARTFUN1:sch 57
s57{
F1()
-> set ,
F2()
-> set ,
P1[
set ,
set ] } :
ex
f being
PartFunc of
F1(),
F2() st
( ( for
x being
set holds
(
x in dom f iff (
x in F1() & ex
y being
set st
P1[
x,
y] ) ) ) & ( for
x being
set st
x in dom f holds
P1[
x,
f . x] ) )
provided
E26:
for
x,
y being
set st
x in F1() &
P1[
x,
y] holds
y in F2()
and E31:
for
x,
y1,
y2 being
set st
x in F1() &
P1[
x,
y1] &
P1[
x,
y2] holds
y1 = y2
scheme :: PARTFUN1:sch 65
s65{
F1()
-> set ,
F2()
-> set ,
F3()
-> set ,
P1[
set ,
set ,
set ] } :
ex
f being
PartFunc of
[:F1(),F2():],
F3() st
( ( for
x,
y being
set holds
(
[x,y] in dom f iff (
x in F1() &
y in F2() & ex
z being
set st
P1[
x,
y,
z] ) ) ) & ( for
x,
y being
set st
[x,y] in dom f holds
P1[
x,
y,
f . [x,y]] ) )
provided
E26:
for
x,
y,
z being
set st
x in F1() &
y in F2() &
P1[
x,
y,
z] holds
z in F3()
and E31:
for
x,
y,
z1,
z2 being
set st
x in F1() &
y in F2() &
P1[
x,
y,
z1] &
P1[
x,
y,
z2] holds
z1 = z2
scheme :: PARTFUN1:sch 67
s67{
F1()
-> set ,
F2()
-> set ,
F3()
-> set ,
F4(
set ,
set )
-> set ,
P1[
set ,
set ] } :
ex
f being
PartFunc of
[:F1(),F2():],
F3() st
( ( for
x,
y being
set holds
(
[x,y] in dom f iff (
x in F1() &
y in F2() &
P1[
x,
y] ) ) ) & ( for
x,
y being
set st
[x,y] in dom f holds
f . [x,y] = F4(
x,
y) ) )
provided
E26:
for
x,
y being
set st
P1[
x,
y] holds
F4(
x,
y)
in F3()
theorem Th36: :: PARTFUN1:36
theorem Th37: :: PARTFUN1:37
theorem Th38: :: PARTFUN1:38
theorem Th39: :: PARTFUN1:39
theorem Th40: :: PARTFUN1:40
canceled;
theorem Th41: :: PARTFUN1:41
canceled;
theorem Th42: :: PARTFUN1:42
canceled;
theorem Th43: :: PARTFUN1:43
theorem Th44: :: PARTFUN1:44
theorem Th45: :: PARTFUN1:45
theorem Th46: :: PARTFUN1:46
theorem Th47: :: PARTFUN1:47
theorem Th48: :: PARTFUN1:48
canceled;
theorem Th49: :: PARTFUN1:49
theorem Th50: :: PARTFUN1:50
canceled;
theorem Th51: :: PARTFUN1:51
theorem Th52: :: PARTFUN1:52
canceled;
theorem Th53: :: PARTFUN1:53
theorem Th54: :: PARTFUN1:54
theorem Th55: :: PARTFUN1:55
theorem Th56: :: PARTFUN1:56
theorem Th57: :: PARTFUN1:57
theorem Th58: :: PARTFUN1:58
theorem Th59: :: PARTFUN1:59
theorem Th60: :: PARTFUN1:60
theorem Th61: :: PARTFUN1:61
theorem Th62: :: PARTFUN1:62
theorem Th63: :: PARTFUN1:63
theorem Th64: :: PARTFUN1:64
theorem Th65: :: PARTFUN1:65
theorem Th66: :: PARTFUN1:66
theorem Th67: :: PARTFUN1:67
theorem Th68: :: PARTFUN1:68
theorem Th69: :: PARTFUN1:69
theorem Th70: :: PARTFUN1:70
theorem Th71: :: PARTFUN1:71
theorem Th72: :: PARTFUN1:72
theorem Th73: :: PARTFUN1:73
theorem Th74: :: PARTFUN1:74
:: deftheorem Def1 PARTFUN1:def 1 :
canceled;
:: deftheorem Def2 PARTFUN1:def 2 :
canceled;
:: deftheorem Def3 defines <: PARTFUN1:def 3 :
theorem Th75: :: PARTFUN1:75
canceled;
theorem Th76: :: PARTFUN1:76
theorem Th77: :: PARTFUN1:77
theorem Th78: :: PARTFUN1:78
theorem Th79: :: PARTFUN1:79
theorem Th80: :: PARTFUN1:80
theorem Th81: :: PARTFUN1:81
theorem Th82: :: PARTFUN1:82
theorem Th83: :: PARTFUN1:83
theorem Th84: :: PARTFUN1:84
theorem Th85: :: PARTFUN1:85
theorem Th86: :: PARTFUN1:86
theorem Th87: :: PARTFUN1:87
theorem Th88: :: PARTFUN1:88
canceled;
theorem Th89: :: PARTFUN1:89
canceled;
theorem Th90: :: PARTFUN1:90
canceled;
theorem Th91: :: PARTFUN1:91
theorem Th92: :: PARTFUN1:92
theorem Th93: :: PARTFUN1:93
theorem Th94: :: PARTFUN1:94
theorem Th95: :: PARTFUN1:95
theorem Th96: :: PARTFUN1:96
theorem Th97: :: PARTFUN1:97
:: deftheorem Def4 defines total PARTFUN1:def 4 :
theorem Th98: :: PARTFUN1:98
canceled;
theorem Th99: :: PARTFUN1:99
theorem Th100: :: PARTFUN1:100
canceled;
theorem Th101: :: PARTFUN1:101
canceled;
theorem Th102: :: PARTFUN1:102
canceled;
theorem Th103: :: PARTFUN1:103
canceled;
theorem Th104: :: PARTFUN1:104
canceled;
theorem Th105: :: PARTFUN1:105
canceled;
theorem Th106: :: PARTFUN1:106
canceled;
theorem Th107: :: PARTFUN1:107
canceled;
theorem Th108: :: PARTFUN1:108
canceled;
theorem Th109: :: PARTFUN1:109
canceled;
theorem Th110: :: PARTFUN1:110
canceled;
theorem Th111: :: PARTFUN1:111
canceled;
theorem Th112: :: PARTFUN1:112
theorem Th113: :: PARTFUN1:113
theorem Th114: :: PARTFUN1:114
theorem Th115: :: PARTFUN1:115
theorem Th116: :: PARTFUN1:116
theorem Th117: :: PARTFUN1:117
:: deftheorem Def5 defines PFuncs PARTFUN1:def 5 :
theorem Th118: :: PARTFUN1:118
canceled;
theorem Th119: :: PARTFUN1:119
theorem Th120: :: PARTFUN1:120
theorem Th121: :: PARTFUN1:121
theorem Th122: :: PARTFUN1:122
theorem Th123: :: PARTFUN1:123
theorem Th124: :: PARTFUN1:124
canceled;
theorem Th125: :: PARTFUN1:125
theorem Th126: :: PARTFUN1:126
theorem Th127: :: PARTFUN1:127
theorem Th128: :: PARTFUN1:128
:: deftheorem Def6 defines tolerates PARTFUN1:def 6 :
theorem Th129: :: PARTFUN1:129
canceled;
theorem Th130: :: PARTFUN1:130
theorem Th131: :: PARTFUN1:131
theorem Th132: :: PARTFUN1:132
theorem Th133: :: PARTFUN1:133
canceled;
theorem Th134: :: PARTFUN1:134
canceled;
theorem Th135: :: PARTFUN1:135
theorem Th136: :: PARTFUN1:136
theorem Th137: :: PARTFUN1:137
canceled;
theorem Th138: :: PARTFUN1:138
theorem Th139: :: PARTFUN1:139
theorem Th140: :: PARTFUN1:140
theorem Th141: :: PARTFUN1:141
theorem Th142: :: PARTFUN1:142
theorem Th143: :: PARTFUN1:143
theorem Th144: :: PARTFUN1:144
theorem Th145: :: PARTFUN1:145
theorem Th146: :: PARTFUN1:146
theorem Th147: :: PARTFUN1:147
theorem Th148: :: PARTFUN1:148
theorem Th149: :: PARTFUN1:149
canceled;
theorem Th150: :: PARTFUN1:150
canceled;
theorem Th151: :: PARTFUN1:151
canceled;
theorem Th152: :: PARTFUN1:152
canceled;
theorem Th153: :: PARTFUN1:153
canceled;
theorem Th154: :: PARTFUN1:154
canceled;
theorem Th155: :: PARTFUN1:155
canceled;
theorem Th156: :: PARTFUN1:156
canceled;
theorem Th157: :: PARTFUN1:157
canceled;
theorem Th158: :: PARTFUN1:158
theorem Th159: :: PARTFUN1:159
canceled;
theorem Th160: :: PARTFUN1:160
canceled;
theorem Th161: :: PARTFUN1:161
canceled;
theorem Th162: :: PARTFUN1:162
:: deftheorem Def7 defines TotFuncs PARTFUN1:def 7 :
theorem Th163: :: PARTFUN1:163
canceled;
theorem Th164: :: PARTFUN1:164
canceled;
theorem Th165: :: PARTFUN1:165
canceled;
theorem Th166: :: PARTFUN1:166
canceled;
theorem Th167: :: PARTFUN1:167
canceled;
theorem Th168: :: PARTFUN1:168
theorem Th169: :: PARTFUN1:169
theorem Th170: :: PARTFUN1:170
canceled;
theorem Th171: :: PARTFUN1:171
theorem Th172: :: PARTFUN1:172
theorem Th173: :: PARTFUN1:173
canceled;
theorem Th174: :: PARTFUN1:174
theorem Th175: :: PARTFUN1:175
theorem Th176: :: PARTFUN1:176
theorem Th177: :: PARTFUN1:177
canceled;
theorem Th178: :: PARTFUN1:178
canceled;
theorem Th179: :: PARTFUN1:179
canceled;
theorem Th180: :: PARTFUN1:180
canceled;
theorem Th181: :: PARTFUN1:181
canceled;
theorem Th182: :: PARTFUN1:182
canceled;
theorem Th183: :: PARTFUN1:183
canceled;
theorem Th184: :: PARTFUN1:184
canceled;
theorem Th185: :: PARTFUN1:185
theorem Th186: :: PARTFUN1:186
Lemma116:
for X being set
for R being Relation of X st R = id X holds
R is total
Lemma117:
for X being set
for R being Relation st R = id X holds
( R is reflexive & R is symmetric & R is antisymmetric & R is transitive )
Lemma118:
for X being set holds id X is Relation of X