:: Relations and Their Basic Properties
:: by Edmund Woronowicz
::
:: Received March 15, 1989
:: Copyright (c) 1990 Association of Mizar Users
:: deftheorem Def1 defines Relation-like RELAT_1:def 1 :
theorem :: RELAT_1:1
canceled;
theorem :: RELAT_1:2
canceled;
theorem Th3: :: RELAT_1:3
theorem Th4: :: RELAT_1:4
theorem Th5: :: RELAT_1:5
theorem Th6: :: RELAT_1:6
Lm1:
for x, y, X being set st [x,y] in X holds
( x in union (union X) & y in union (union X) )
:: deftheorem Def2 defines = RELAT_1:def 2 :
:: deftheorem Def3 defines c= RELAT_1:def 3 :
theorem :: RELAT_1:7
canceled;
theorem :: RELAT_1:8
canceled;
theorem :: RELAT_1:9
theorem :: RELAT_1:10
:: deftheorem Def4 defines proj1 RELAT_1:def 4 :
for
R,
b2 being
set holds
(
b2 = proj1 R iff for
x being
set holds
(
x in b2 iff ex
y being
set st
[x,y] in R ) );
theorem :: RELAT_1:11
canceled;
theorem :: RELAT_1:12
canceled;
theorem Th13: :: RELAT_1:13
theorem Th14: :: RELAT_1:14
theorem :: RELAT_1:15
:: deftheorem Def5 defines proj2 RELAT_1:def 5 :
for
R,
b2 being
set holds
(
b2 = proj2 R iff for
y being
set holds
(
y in b2 iff ex
x being
set st
[x,y] in R ) );
theorem :: RELAT_1:16
canceled;
theorem :: RELAT_1:17
canceled;
theorem :: RELAT_1:18
theorem :: RELAT_1:19
theorem :: RELAT_1:20
theorem Th21: :: RELAT_1:21
theorem :: RELAT_1:22
theorem Th23: :: RELAT_1:23
theorem :: RELAT_1:24
theorem Th25: :: RELAT_1:25
theorem Th26: :: RELAT_1:26
theorem Th27: :: RELAT_1:27
theorem :: RELAT_1:28
:: deftheorem defines field RELAT_1:def 6 :
theorem :: RELAT_1:29
canceled;
theorem :: RELAT_1:30
theorem :: RELAT_1:31
theorem :: RELAT_1:32
theorem :: RELAT_1:33
theorem :: RELAT_1:34
definition
let R be
Relation;
func R ~ -> Relation means :
Def7:
:: RELAT_1:def 7
for
x,
y being
set holds
(
[x,y] in it iff
[y,x] in R );
existence
ex b1 being Relation st
for x, y being set holds
( [x,y] in b1 iff [y,x] in R )
uniqueness
for b1, b2 being Relation st ( for x, y being set holds
( [x,y] in b1 iff [y,x] in R ) ) & ( for x, y being set holds
( [x,y] in b2 iff [y,x] in R ) ) holds
b1 = b2
involutiveness
for b1, b2 being Relation st ( for x, y being set holds
( [x,y] in b1 iff [y,x] in b2 ) ) holds
for x, y being set holds
( [x,y] in b2 iff [y,x] in b1 )
;
end;
:: deftheorem Def7 defines ~ RELAT_1:def 7 :
theorem :: RELAT_1:35
canceled;
theorem :: RELAT_1:36
canceled;
theorem Th37: :: RELAT_1:37
theorem :: RELAT_1:38
theorem :: RELAT_1:39
theorem :: RELAT_1:40
theorem :: RELAT_1:41
definition
let P,
R be
Relation;
func P * R -> Relation means :
Def8:
:: RELAT_1:def 8
for
x,
y being
set holds
(
[x,y] in it iff ex
z being
set st
(
[x,z] in P &
[z,y] in R ) );
existence
ex b1 being Relation st
for x, y being set holds
( [x,y] in b1 iff ex z being set st
( [x,z] in P & [z,y] in R ) )
uniqueness
for b1, b2 being Relation st ( for x, y being set holds
( [x,y] in b1 iff ex z being set st
( [x,z] in P & [z,y] in R ) ) ) & ( for x, y being set holds
( [x,y] in b2 iff ex z being set st
( [x,z] in P & [z,y] in R ) ) ) holds
b1 = b2
end;
:: deftheorem Def8 defines * RELAT_1:def 8 :
for
P,
R,
b3 being
Relation holds
(
b3 = P * R iff for
x,
y being
set holds
(
[x,y] in b3 iff ex
z being
set st
(
[x,z] in P &
[z,y] in R ) ) );
theorem :: RELAT_1:42
canceled;
theorem :: RELAT_1:43
canceled;
theorem Th44: :: RELAT_1:44
theorem Th45: :: RELAT_1:45
theorem :: RELAT_1:46
theorem :: RELAT_1:47
theorem Th48: :: RELAT_1:48
theorem Th49: :: RELAT_1:49
theorem :: RELAT_1:50
theorem :: RELAT_1:51
theorem :: RELAT_1:52
theorem :: RELAT_1:53
theorem :: RELAT_1:54
theorem Th55: :: RELAT_1:55
theorem Th56: :: RELAT_1:56
theorem :: RELAT_1:57
canceled;
theorem :: RELAT_1:58
canceled;
theorem :: RELAT_1:59
canceled;
theorem Th60: :: RELAT_1:60
theorem :: RELAT_1:61
canceled;
theorem Th62: :: RELAT_1:62
theorem :: RELAT_1:63
theorem Th64: :: RELAT_1:64
theorem :: RELAT_1:65
theorem :: RELAT_1:66
theorem :: RELAT_1:67
:: deftheorem Def9 defines non-empty RELAT_1:def 9 :
:: deftheorem Def10 defines id RELAT_1:def 10 :
for
X being
set for
b2 being
Relation holds
(
b2 = id X iff for
x,
y being
set holds
(
[x,y] in b2 iff (
x in X &
x = y ) ) );
theorem :: RELAT_1:68
canceled;
theorem :: RELAT_1:69
canceled;
theorem :: RELAT_1:70
canceled;
theorem Th71: :: RELAT_1:71
theorem :: RELAT_1:72
theorem :: RELAT_1:73
theorem Th74: :: RELAT_1:74
theorem Th75: :: RELAT_1:75
theorem Th76: :: RELAT_1:76
theorem Th77: :: RELAT_1:77
theorem :: RELAT_1:78
theorem Th79: :: RELAT_1:79
theorem :: RELAT_1:80
theorem :: RELAT_1:81
theorem :: RELAT_1:82
definition
let R be
Relation;
let X be
set ;
func R | X -> Relation means :
Def11:
:: RELAT_1:def 11
for
x,
y being
set holds
(
[x,y] in it iff (
x in X &
[x,y] in R ) );
existence
ex b1 being Relation st
for x, y being set holds
( [x,y] in b1 iff ( x in X & [x,y] in R ) )
uniqueness
for b1, b2 being Relation st ( for x, y being set holds
( [x,y] in b1 iff ( x in X & [x,y] in R ) ) ) & ( for x, y being set holds
( [x,y] in b2 iff ( x in X & [x,y] in R ) ) ) holds
b1 = b2
end;
:: deftheorem Def11 defines | RELAT_1:def 11 :
theorem :: RELAT_1:83
canceled;
theorem :: RELAT_1:84
canceled;
theorem :: RELAT_1:85
canceled;
theorem Th86: :: RELAT_1:86
theorem :: RELAT_1:87
theorem Th88: :: RELAT_1:88
theorem :: RELAT_1:89
theorem Th90: :: RELAT_1:90
theorem :: RELAT_1:91
theorem :: RELAT_1:92
theorem :: RELAT_1:93
theorem :: RELAT_1:94
theorem :: RELAT_1:95
theorem Th96: :: RELAT_1:96
theorem Th97: :: RELAT_1:97
theorem :: RELAT_1:98
theorem Th99: :: RELAT_1:99
theorem Th100: :: RELAT_1:100
theorem :: RELAT_1:101
theorem :: RELAT_1:102
theorem :: RELAT_1:103
theorem Th104: :: RELAT_1:104
theorem Th105: :: RELAT_1:105
theorem Th106: :: RELAT_1:106
theorem Th107: :: RELAT_1:107
theorem :: RELAT_1:108
theorem :: RELAT_1:109
theorem :: RELAT_1:110
theorem :: RELAT_1:111
theorem :: RELAT_1:112
definition
let Y be
set ;
let R be
Relation;
func Y | R -> Relation means :
Def12:
:: RELAT_1:def 12
for
x,
y being
set holds
(
[x,y] in it iff (
y in Y &
[x,y] in R ) );
existence
ex b1 being Relation st
for x, y being set holds
( [x,y] in b1 iff ( y in Y & [x,y] in R ) )
uniqueness
for b1, b2 being Relation st ( for x, y being set holds
( [x,y] in b1 iff ( y in Y & [x,y] in R ) ) ) & ( for x, y being set holds
( [x,y] in b2 iff ( y in Y & [x,y] in R ) ) ) holds
b1 = b2
end;
:: deftheorem Def12 defines | RELAT_1:def 12 :
for
Y being
set for
R,
b3 being
Relation holds
(
b3 = Y | R iff for
x,
y being
set holds
(
[x,y] in b3 iff (
y in Y &
[x,y] in R ) ) );
theorem :: RELAT_1:113
canceled;
theorem :: RELAT_1:114
canceled;
theorem Th115: :: RELAT_1:115
theorem Th116: :: RELAT_1:116
theorem Th117: :: RELAT_1:117
theorem Th118: :: RELAT_1:118
theorem Th119: :: RELAT_1:119
theorem :: RELAT_1:120
theorem :: RELAT_1:121
theorem :: RELAT_1:122
theorem :: RELAT_1:123
theorem Th124: :: RELAT_1:124
theorem Th125: :: RELAT_1:125
theorem :: RELAT_1:126
theorem Th127: :: RELAT_1:127
theorem :: RELAT_1:128
theorem :: RELAT_1:129
theorem :: RELAT_1:130
theorem Th131: :: RELAT_1:131
theorem Th132: :: RELAT_1:132
theorem :: RELAT_1:133
theorem :: RELAT_1:134
theorem :: RELAT_1:135
theorem :: RELAT_1:136
theorem :: RELAT_1:137
theorem :: RELAT_1:138
theorem :: RELAT_1:139
theorem :: RELAT_1:140
:: deftheorem Def13 defines .: RELAT_1:def 13 :
for
R being
Relation for
X,
b3 being
set holds
(
b3 = R .: X iff for
y being
set holds
(
y in b3 iff ex
x being
set st
(
[x,y] in R &
x in X ) ) );
theorem :: RELAT_1:141
canceled;
theorem :: RELAT_1:142
canceled;
theorem Th143: :: RELAT_1:143
theorem Th144: :: RELAT_1:144
theorem :: RELAT_1:145
theorem Th146: :: RELAT_1:146
theorem :: RELAT_1:147
theorem :: RELAT_1:148
theorem :: RELAT_1:149
theorem :: RELAT_1:150
theorem :: RELAT_1:151
theorem :: RELAT_1:152
theorem :: RELAT_1:153
theorem :: RELAT_1:154
theorem :: RELAT_1:155
theorem Th156: :: RELAT_1:156
theorem Th157: :: RELAT_1:157
theorem :: RELAT_1:158
theorem :: RELAT_1:159
theorem Th160: :: RELAT_1:160
theorem :: RELAT_1:161
theorem Th162: :: RELAT_1:162
theorem :: RELAT_1:163
:: deftheorem Def14 defines " RELAT_1:def 14 :
for
R being
Relation for
Y,
b3 being
set holds
(
b3 = R " Y iff for
x being
set holds
(
x in b3 iff ex
y being
set st
(
[x,y] in R &
y in Y ) ) );
theorem :: RELAT_1:164
canceled;
theorem :: RELAT_1:165
canceled;
theorem Th166: :: RELAT_1:166
theorem Th167: :: RELAT_1:167
theorem :: RELAT_1:168
theorem Th169: :: RELAT_1:169
theorem :: RELAT_1:170
theorem :: RELAT_1:171
theorem :: RELAT_1:172
theorem :: RELAT_1:173
theorem :: RELAT_1:174
theorem :: RELAT_1:175
theorem :: RELAT_1:176
theorem :: RELAT_1:177
theorem Th178: :: RELAT_1:178
theorem Th179: :: RELAT_1:179
theorem :: RELAT_1:180
theorem :: RELAT_1:181
theorem Th182: :: RELAT_1:182
theorem :: RELAT_1:183
:: deftheorem Def15 defines empty-yielding RELAT_1:def 15 :
theorem :: RELAT_1:184
theorem :: RELAT_1:185
theorem :: RELAT_1:186
theorem :: RELAT_1:187
theorem :: RELAT_1:188
theorem :: RELAT_1:189
theorem :: RELAT_1:190
:: deftheorem defines Im RELAT_1:def 16 :
theorem :: RELAT_1:191
theorem :: RELAT_1:192
theorem :: RELAT_1:193
theorem :: RELAT_1:194
theorem :: RELAT_1:195
theorem :: RELAT_1:196
theorem :: RELAT_1:197
theorem :: RELAT_1:198
theorem :: RELAT_1:199
:: deftheorem defines Coim RELAT_1:def 17 :
theorem :: RELAT_1:200
theorem :: RELAT_1:201