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Theorem domss2 7673
Description: A corollary of disjenex 7672. If  F is an injection from  A to  B then  G is a right inverse of  F from  B to a superset of  A. (Contributed by Mario Carneiro, 7-Feb-2015.) (Revised by Mario Carneiro, 24-Jun-2015.)
Hypothesis
Ref Expression
domss2.1  |-  G  =  `' ( F  u.  ( 1st  |`  ( ( B  \  ran  F )  X.  { ~P U. ran  A } ) ) )
Assertion
Ref Expression
domss2  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  ( G : B -1-1-onto-> ran  G  /\  A  C_ 
ran  G  /\  ( G  o.  F )  =  (  _I  |`  A ) ) )

Proof of Theorem domss2
StepHypRef Expression
1 f1f1orn 5825 . . . . . . . 8  |-  ( F : A -1-1-> B  ->  F : A -1-1-onto-> ran  F )
213ad2ant1 1017 . . . . . . 7  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  F : A
-1-1-onto-> ran  F )
3 simp2 997 . . . . . . . . . 10  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  A  e.  V )
4 rnexg 6713 . . . . . . . . . 10  |-  ( A  e.  V  ->  ran  A  e.  _V )
53, 4syl 16 . . . . . . . . 9  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  ran  A  e. 
_V )
6 uniexg 6579 . . . . . . . . 9  |-  ( ran 
A  e.  _V  ->  U.
ran  A  e.  _V )
7 pwexg 4631 . . . . . . . . 9  |-  ( U. ran  A  e.  _V  ->  ~P
U. ran  A  e.  _V )
85, 6, 73syl 20 . . . . . . . 8  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  ~P U. ran  A  e.  _V )
9 1stconst 6868 . . . . . . . 8  |-  ( ~P
U. ran  A  e.  _V  ->  ( 1st  |`  (
( B  \  ran  F )  X.  { ~P U.
ran  A } ) ) : ( ( B  \  ran  F
)  X.  { ~P U.
ran  A } ) -1-1-onto-> ( B  \  ran  F
) )
108, 9syl 16 . . . . . . 7  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  ( 1st  |`  ( ( B  \  ran  F )  X.  { ~P U. ran  A }
) ) : ( ( B  \  ran  F )  X.  { ~P U.
ran  A } ) -1-1-onto-> ( B  \  ran  F
) )
11 difexg 4595 . . . . . . . . . 10  |-  ( B  e.  W  ->  ( B  \  ran  F )  e.  _V )
12113ad2ant3 1019 . . . . . . . . 9  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  ( B  \  ran  F )  e. 
_V )
13 disjen 7671 . . . . . . . . 9  |-  ( ( A  e.  V  /\  ( B  \  ran  F
)  e.  _V )  ->  ( ( A  i^i  ( ( B  \  ran  F )  X.  { ~P U. ran  A }
) )  =  (/)  /\  ( ( B  \  ran  F )  X.  { ~P U. ran  A }
)  ~~  ( B  \  ran  F ) ) )
143, 12, 13syl2anc 661 . . . . . . . 8  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  ( ( A  i^i  ( ( B 
\  ran  F )  X.  { ~P U. ran  A } ) )  =  (/)  /\  ( ( B 
\  ran  F )  X.  { ~P U. ran  A } )  ~~  ( B  \  ran  F ) ) )
1514simpld 459 . . . . . . 7  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  ( A  i^i  ( ( B  \  ran  F )  X.  { ~P U. ran  A }
) )  =  (/) )
16 disjdif 3899 . . . . . . . 8  |-  ( ran 
F  i^i  ( B  \  ran  F ) )  =  (/)
1716a1i 11 . . . . . . 7  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  ( ran  F  i^i  ( B  \  ran  F ) )  =  (/) )
18 f1oun 5833 . . . . . . 7  |-  ( ( ( F : A -1-1-onto-> ran  F  /\  ( 1st  |`  (
( B  \  ran  F )  X.  { ~P U.
ran  A } ) ) : ( ( B  \  ran  F
)  X.  { ~P U.
ran  A } ) -1-1-onto-> ( B  \  ran  F
) )  /\  (
( A  i^i  (
( B  \  ran  F )  X.  { ~P U.
ran  A } ) )  =  (/)  /\  ( ran  F  i^i  ( B 
\  ran  F )
)  =  (/) ) )  ->  ( F  u.  ( 1st  |`  ( ( B  \  ran  F )  X.  { ~P U. ran  A } ) ) ) : ( A  u.  ( ( B 
\  ran  F )  X.  { ~P U. ran  A } ) ) -1-1-onto-> ( ran 
F  u.  ( B 
\  ran  F )
) )
192, 10, 15, 17, 18syl22anc 1229 . . . . . 6  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  ( F  u.  ( 1st  |`  (
( B  \  ran  F )  X.  { ~P U.
ran  A } ) ) ) : ( A  u.  ( ( B  \  ran  F
)  X.  { ~P U.
ran  A } ) ) -1-1-onto-> ( ran  F  u.  ( B  \  ran  F
) ) )
20 undif2 3903 . . . . . . . 8  |-  ( ran 
F  u.  ( B 
\  ran  F )
)  =  ( ran 
F  u.  B )
21 f1f 5779 . . . . . . . . . . 11  |-  ( F : A -1-1-> B  ->  F : A --> B )
22213ad2ant1 1017 . . . . . . . . . 10  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  F : A
--> B )
23 frn 5735 . . . . . . . . . 10  |-  ( F : A --> B  ->  ran  F  C_  B )
2422, 23syl 16 . . . . . . . . 9  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  ran  F  C_  B )
25 ssequn1 3674 . . . . . . . . 9  |-  ( ran 
F  C_  B  <->  ( ran  F  u.  B )  =  B )
2624, 25sylib 196 . . . . . . . 8  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  ( ran  F  u.  B )  =  B )
2720, 26syl5eq 2520 . . . . . . 7  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  ( ran  F  u.  ( B  \  ran  F ) )  =  B )
28 f1oeq3 5807 . . . . . . 7  |-  ( ( ran  F  u.  ( B  \  ran  F ) )  =  B  -> 
( ( F  u.  ( 1st  |`  ( ( B  \  ran  F )  X.  { ~P U. ran  A } ) ) ) : ( A  u.  ( ( B 
\  ran  F )  X.  { ~P U. ran  A } ) ) -1-1-onto-> ( ran 
F  u.  ( B 
\  ran  F )
)  <->  ( F  u.  ( 1st  |`  ( ( B  \  ran  F )  X.  { ~P U. ran  A } ) ) ) : ( A  u.  ( ( B 
\  ran  F )  X.  { ~P U. ran  A } ) ) -1-1-onto-> B ) )
2927, 28syl 16 . . . . . 6  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  ( ( F  u.  ( 1st  |`  ( ( B  \  ran  F )  X.  { ~P U. ran  A }
) ) ) : ( A  u.  (
( B  \  ran  F )  X.  { ~P U.
ran  A } ) ) -1-1-onto-> ( ran  F  u.  ( B  \  ran  F
) )  <->  ( F  u.  ( 1st  |`  (
( B  \  ran  F )  X.  { ~P U.
ran  A } ) ) ) : ( A  u.  ( ( B  \  ran  F
)  X.  { ~P U.
ran  A } ) ) -1-1-onto-> B ) )
3019, 29mpbid 210 . . . . 5  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  ( F  u.  ( 1st  |`  (
( B  \  ran  F )  X.  { ~P U.
ran  A } ) ) ) : ( A  u.  ( ( B  \  ran  F
)  X.  { ~P U.
ran  A } ) ) -1-1-onto-> B )
31 f1ocnv 5826 . . . . 5  |-  ( ( F  u.  ( 1st  |`  ( ( B  \  ran  F )  X.  { ~P U. ran  A }
) ) ) : ( A  u.  (
( B  \  ran  F )  X.  { ~P U.
ran  A } ) ) -1-1-onto-> B  ->  `' ( F  u.  ( 1st  |`  ( ( B  \  ran  F )  X.  { ~P U. ran  A }
) ) ) : B -1-1-onto-> ( A  u.  (
( B  \  ran  F )  X.  { ~P U.
ran  A } ) ) )
3230, 31syl 16 . . . 4  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  `' ( F  u.  ( 1st  |`  ( ( B  \  ran  F )  X.  { ~P U. ran  A }
) ) ) : B -1-1-onto-> ( A  u.  (
( B  \  ran  F )  X.  { ~P U.
ran  A } ) ) )
33 domss2.1 . . . . 5  |-  G  =  `' ( F  u.  ( 1st  |`  ( ( B  \  ran  F )  X.  { ~P U. ran  A } ) ) )
34 f1oeq1 5805 . . . . 5  |-  ( G  =  `' ( F  u.  ( 1st  |`  (
( B  \  ran  F )  X.  { ~P U.
ran  A } ) ) )  ->  ( G : B -1-1-onto-> ( A  u.  (
( B  \  ran  F )  X.  { ~P U.
ran  A } ) )  <->  `' ( F  u.  ( 1st  |`  ( ( B  \  ran  F )  X.  { ~P U. ran  A } ) ) ) : B -1-1-onto-> ( A  u.  ( ( B 
\  ran  F )  X.  { ~P U. ran  A } ) ) ) )
3533, 34ax-mp 5 . . . 4  |-  ( G : B -1-1-onto-> ( A  u.  (
( B  \  ran  F )  X.  { ~P U.
ran  A } ) )  <->  `' ( F  u.  ( 1st  |`  ( ( B  \  ran  F )  X.  { ~P U. ran  A } ) ) ) : B -1-1-onto-> ( A  u.  ( ( B 
\  ran  F )  X.  { ~P U. ran  A } ) ) )
3632, 35sylibr 212 . . 3  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  G : B
-1-1-onto-> ( A  u.  (
( B  \  ran  F )  X.  { ~P U.
ran  A } ) ) )
37 f1ofo 5821 . . . . 5  |-  ( G : B -1-1-onto-> ( A  u.  (
( B  \  ran  F )  X.  { ~P U.
ran  A } ) )  ->  G : B -onto-> ( A  u.  ( ( B  \  ran  F )  X.  { ~P U. ran  A }
) ) )
38 forn 5796 . . . . 5  |-  ( G : B -onto-> ( A  u.  ( ( B 
\  ran  F )  X.  { ~P U. ran  A } ) )  ->  ran  G  =  ( A  u.  ( ( B 
\  ran  F )  X.  { ~P U. ran  A } ) ) )
3936, 37, 383syl 20 . . . 4  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  ran  G  =  ( A  u.  (
( B  \  ran  F )  X.  { ~P U.
ran  A } ) ) )
40 f1oeq3 5807 . . . 4  |-  ( ran 
G  =  ( A  u.  ( ( B 
\  ran  F )  X.  { ~P U. ran  A } ) )  -> 
( G : B -1-1-onto-> ran  G  <-> 
G : B -1-1-onto-> ( A  u.  ( ( B 
\  ran  F )  X.  { ~P U. ran  A } ) ) ) )
4139, 40syl 16 . . 3  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  ( G : B -1-1-onto-> ran  G  <->  G : B
-1-1-onto-> ( A  u.  (
( B  \  ran  F )  X.  { ~P U.
ran  A } ) ) ) )
4236, 41mpbird 232 . 2  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  G : B
-1-1-onto-> ran  G )
43 ssun1 3667 . . 3  |-  A  C_  ( A  u.  (
( B  \  ran  F )  X.  { ~P U.
ran  A } ) )
4443, 39syl5sseqr 3553 . 2  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  A  C_  ran  G )
45 ssid 3523 . . . 4  |-  ran  F  C_ 
ran  F
46 cores 5508 . . . 4  |-  ( ran 
F  C_  ran  F  -> 
( ( G  |`  ran  F )  o.  F
)  =  ( G  o.  F ) )
4745, 46ax-mp 5 . . 3  |-  ( ( G  |`  ran  F )  o.  F )  =  ( G  o.  F
)
48 dmres 5292 . . . . . . . . 9  |-  dom  ( `' ( 1st  |`  (
( B  \  ran  F )  X.  { ~P U.
ran  A } ) )  |`  ran  F )  =  ( ran  F  i^i  dom  `' ( 1st  |`  ( ( B  \  ran  F )  X.  { ~P U. ran  A }
) ) )
49 f1ocnv 5826 . . . . . . . . . . . 12  |-  ( ( 1st  |`  ( ( B  \  ran  F )  X.  { ~P U. ran  A } ) ) : ( ( B 
\  ran  F )  X.  { ~P U. ran  A } ) -1-1-onto-> ( B  \  ran  F )  ->  `' ( 1st  |`  ( ( B 
\  ran  F )  X.  { ~P U. ran  A } ) ) : ( B  \  ran  F ) -1-1-onto-> ( ( B  \  ran  F )  X.  { ~P U. ran  A }
) )
50 f1odm 5818 . . . . . . . . . . . 12  |-  ( `' ( 1st  |`  (
( B  \  ran  F )  X.  { ~P U.
ran  A } ) ) : ( B 
\  ran  F ) -1-1-onto-> (
( B  \  ran  F )  X.  { ~P U.
ran  A } )  ->  dom  `' ( 1st  |`  ( ( B 
\  ran  F )  X.  { ~P U. ran  A } ) )  =  ( B  \  ran  F ) )
5110, 49, 503syl 20 . . . . . . . . . . 11  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  dom  `' ( 1st  |`  ( ( B  \  ran  F )  X.  { ~P U. ran  A } ) )  =  ( B  \  ran  F ) )
5251ineq2d 3700 . . . . . . . . . 10  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  ( ran  F  i^i  dom  `' ( 1st  |`  ( ( B 
\  ran  F )  X.  { ~P U. ran  A } ) ) )  =  ( ran  F  i^i  ( B  \  ran  F ) ) )
5352, 16syl6eq 2524 . . . . . . . . 9  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  ( ran  F  i^i  dom  `' ( 1st  |`  ( ( B 
\  ran  F )  X.  { ~P U. ran  A } ) ) )  =  (/) )
5448, 53syl5eq 2520 . . . . . . . 8  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  dom  ( `' ( 1st  |`  (
( B  \  ran  F )  X.  { ~P U.
ran  A } ) )  |`  ran  F )  =  (/) )
55 relres 5299 . . . . . . . . 9  |-  Rel  ( `' ( 1st  |`  (
( B  \  ran  F )  X.  { ~P U.
ran  A } ) )  |`  ran  F )
56 reldm0 5218 . . . . . . . . 9  |-  ( Rel  ( `' ( 1st  |`  ( ( B  \  ran  F )  X.  { ~P U. ran  A }
) )  |`  ran  F
)  ->  ( ( `' ( 1st  |`  (
( B  \  ran  F )  X.  { ~P U.
ran  A } ) )  |`  ran  F )  =  (/)  <->  dom  ( `' ( 1st  |`  ( ( B  \  ran  F )  X.  { ~P U. ran  A } ) )  |`  ran  F )  =  (/) ) )
5755, 56ax-mp 5 . . . . . . . 8  |-  ( ( `' ( 1st  |`  (
( B  \  ran  F )  X.  { ~P U.
ran  A } ) )  |`  ran  F )  =  (/)  <->  dom  ( `' ( 1st  |`  ( ( B  \  ran  F )  X.  { ~P U. ran  A } ) )  |`  ran  F )  =  (/) )
5854, 57sylibr 212 . . . . . . 7  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  ( `' ( 1st  |`  ( ( B  \  ran  F )  X.  { ~P U. ran  A } ) )  |`  ran  F )  =  (/) )
5958uneq2d 3658 . . . . . 6  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  ( `' F  u.  ( `' ( 1st  |`  ( ( B  \  ran  F )  X.  { ~P U. ran  A } ) )  |`  ran  F ) )  =  ( `' F  u.  (/) ) )
60 cnvun 5409 . . . . . . . . 9  |-  `' ( F  u.  ( 1st  |`  ( ( B  \  ran  F )  X.  { ~P U. ran  A }
) ) )  =  ( `' F  u.  `' ( 1st  |`  (
( B  \  ran  F )  X.  { ~P U.
ran  A } ) ) )
6133, 60eqtri 2496 . . . . . . . 8  |-  G  =  ( `' F  u.  `' ( 1st  |`  (
( B  \  ran  F )  X.  { ~P U.
ran  A } ) ) )
6261reseq1i 5267 . . . . . . 7  |-  ( G  |`  ran  F )  =  ( ( `' F  u.  `' ( 1st  |`  (
( B  \  ran  F )  X.  { ~P U.
ran  A } ) ) )  |`  ran  F
)
63 resundir 5286 . . . . . . 7  |-  ( ( `' F  u.  `' ( 1st  |`  ( ( B  \  ran  F )  X.  { ~P U. ran  A } ) ) )  |`  ran  F )  =  ( ( `' F  |`  ran  F )  u.  ( `' ( 1st  |`  ( ( B  \  ran  F )  X.  { ~P U. ran  A } ) )  |`  ran  F ) )
64 df-rn 5010 . . . . . . . . . 10  |-  ran  F  =  dom  `' F
6564reseq2i 5268 . . . . . . . . 9  |-  ( `' F  |`  ran  F )  =  ( `' F  |` 
dom  `' F )
66 relcnv 5372 . . . . . . . . . 10  |-  Rel  `' F
67 resdm 5313 . . . . . . . . . 10  |-  ( Rel  `' F  ->  ( `' F  |`  dom  `' F
)  =  `' F
)
6866, 67ax-mp 5 . . . . . . . . 9  |-  ( `' F  |`  dom  `' F
)  =  `' F
6965, 68eqtri 2496 . . . . . . . 8  |-  ( `' F  |`  ran  F )  =  `' F
7069uneq1i 3654 . . . . . . 7  |-  ( ( `' F  |`  ran  F
)  u.  ( `' ( 1st  |`  (
( B  \  ran  F )  X.  { ~P U.
ran  A } ) )  |`  ran  F ) )  =  ( `' F  u.  ( `' ( 1st  |`  (
( B  \  ran  F )  X.  { ~P U.
ran  A } ) )  |`  ran  F ) )
7162, 63, 703eqtrri 2501 . . . . . 6  |-  ( `' F  u.  ( `' ( 1st  |`  (
( B  \  ran  F )  X.  { ~P U.
ran  A } ) )  |`  ran  F ) )  =  ( G  |`  ran  F )
72 un0 3810 . . . . . 6  |-  ( `' F  u.  (/) )  =  `' F
7359, 71, 723eqtr3g 2531 . . . . 5  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  ( G  |` 
ran  F )  =  `' F )
7473coeq1d 5162 . . . 4  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  ( ( G  |`  ran  F )  o.  F )  =  ( `' F  o.  F ) )
75 f1cocnv1 5843 . . . . 5  |-  ( F : A -1-1-> B  -> 
( `' F  o.  F )  =  (  _I  |`  A )
)
76753ad2ant1 1017 . . . 4  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  ( `' F  o.  F )  =  (  _I  |`  A ) )
7774, 76eqtrd 2508 . . 3  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  ( ( G  |`  ran  F )  o.  F )  =  (  _I  |`  A ) )
7847, 77syl5eqr 2522 . 2  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  ( G  o.  F )  =  (  _I  |`  A )
)
7942, 44, 783jca 1176 1  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  ( G : B -1-1-onto-> ran  G  /\  A  C_ 
ran  G  /\  ( G  o.  F )  =  (  _I  |`  A ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   _Vcvv 3113    \ cdif 3473    u. cun 3474    i^i cin 3475    C_ wss 3476   (/)c0 3785   ~Pcpw 4010   {csn 4027   U.cuni 4245   class class class wbr 4447    _I cid 4790    X. cxp 4997   `'ccnv 4998   dom cdm 4999   ran crn 5000    |` cres 5001    o. ccom 5003   Rel wrel 5004   -->wf 5582   -1-1->wf1 5583   -onto->wfo 5584   -1-1-onto->wf1o 5585   1stc1st 6779    ~~ cen 7510
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-1st 6781  df-2nd 6782  df-en 7514
This theorem is referenced by:  domssex2  7674  domssex  7675
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