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Theorem domss2 7669
Description: A corollary of disjenex 7668. 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 5809 . . . . . . . 8  |-  ( F : A -1-1-> B  ->  F : A -1-1-onto-> ran  F )
213ad2ant1 1015 . . . . . . 7  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  F : A
-1-1-onto-> ran  F )
3 simp2 995 . . . . . . . . . 10  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  A  e.  V )
4 rnexg 6705 . . . . . . . . . 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 6570 . . . . . . . . 9  |-  ( ran 
A  e.  _V  ->  U.
ran  A  e.  _V )
7 pwexg 4621 . . . . . . . . 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 6861 . . . . . . . 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 4585 . . . . . . . . . 10  |-  ( B  e.  W  ->  ( B  \  ran  F )  e.  _V )
12113ad2ant3 1017 . . . . . . . . 9  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  ( B  \  ran  F )  e. 
_V )
13 disjen 7667 . . . . . . . . 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 659 . . . . . . . 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 457 . . . . . . 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 3888 . . . . . . . 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 5817 . . . . . . 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 1227 . . . . . 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 3892 . . . . . . . 8  |-  ( ran 
F  u.  ( B 
\  ran  F )
)  =  ( ran 
F  u.  B )
21 f1f 5763 . . . . . . . . . . 11  |-  ( F : A -1-1-> B  ->  F : A --> B )
22213ad2ant1 1015 . . . . . . . . . 10  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  F : A
--> B )
23 frn 5719 . . . . . . . . . 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 3660 . . . . . . . . 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 2507 . . . . . . 7  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  ( ran  F  u.  ( B  \  ran  F ) )  =  B )
28 f1oeq3 5791 . . . . . . 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 5810 . . . . 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 5789 . . . . 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 5805 . . . . 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 5780 . . . . 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 5791 . . . 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 3653 . . 3  |-  A  C_  ( A  u.  (
( B  \  ran  F )  X.  { ~P U.
ran  A } ) )
4443, 39syl5sseqr 3538 . 2  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  A  C_  ran  G )
45 ssid 3508 . . . 4  |-  ran  F  C_ 
ran  F
46 cores 5493 . . . 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 5282 . . . . . . . . 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 5810 . . . . . . . . . . . 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 5802 . . . . . . . . . . . 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 3686 . . . . . . . . . 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 2511 . . . . . . . . 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 2507 . . . . . . . 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 5289 . . . . . . . . 9  |-  Rel  ( `' ( 1st  |`  (
( B  \  ran  F )  X.  { ~P U.
ran  A } ) )  |`  ran  F )
56 reldm0 5209 . . . . . . . . 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 3644 . . . . . 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 5396 . . . . . . . . 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 2483 . . . . . . . 8  |-  G  =  ( `' F  u.  `' ( 1st  |`  (
( B  \  ran  F )  X.  { ~P U.
ran  A } ) ) )
6261reseq1i 5258 . . . . . . 7  |-  ( G  |`  ran  F )  =  ( ( `' F  u.  `' ( 1st  |`  (
( B  \  ran  F )  X.  { ~P U.
ran  A } ) ) )  |`  ran  F
)
63 resundir 5276 . . . . . . 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 4999 . . . . . . . . . 10  |-  ran  F  =  dom  `' F
6564reseq2i 5259 . . . . . . . . 9  |-  ( `' F  |`  ran  F )  =  ( `' F  |` 
dom  `' F )
66 relcnv 5362 . . . . . . . . . 10  |-  Rel  `' F
67 resdm 5303 . . . . . . . . . 10  |-  ( Rel  `' F  ->  ( `' F  |`  dom  `' F
)  =  `' F
)
6866, 67ax-mp 5 . . . . . . . . 9  |-  ( `' F  |`  dom  `' F
)  =  `' F
6965, 68eqtri 2483 . . . . . . . 8  |-  ( `' F  |`  ran  F )  =  `' F
7069uneq1i 3640 . . . . . . 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 2488 . . . . . 6  |-  ( `' F  u.  ( `' ( 1st  |`  (
( B  \  ran  F )  X.  { ~P U.
ran  A } ) )  |`  ran  F ) )  =  ( G  |`  ran  F )
72 un0 3809 . . . . . 6  |-  ( `' F  u.  (/) )  =  `' F
7359, 71, 723eqtr3g 2518 . . . . 5  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  ( G  |` 
ran  F )  =  `' F )
7473coeq1d 5153 . . . 4  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  ( ( G  |`  ran  F )  o.  F )  =  ( `' F  o.  F ) )
75 f1cocnv1 5827 . . . . 5  |-  ( F : A -1-1-> B  -> 
( `' F  o.  F )  =  (  _I  |`  A )
)
76753ad2ant1 1015 . . . 4  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  ( `' F  o.  F )  =  (  _I  |`  A ) )
7774, 76eqtrd 2495 . . 3  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  ( ( G  |`  ran  F )  o.  F )  =  (  _I  |`  A ) )
7847, 77syl5eqr 2509 . 2  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  ( G  o.  F )  =  (  _I  |`  A )
)
7942, 44, 783jca 1174 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823   _Vcvv 3106    \ cdif 3458    u. cun 3459    i^i cin 3460    C_ wss 3461   (/)c0 3783   ~Pcpw 3999   {csn 4016   U.cuni 4235   class class class wbr 4439    _I cid 4779    X. cxp 4986   `'ccnv 4987   dom cdm 4988   ran crn 4989    |` cres 4990    o. ccom 4992   Rel wrel 4993   -->wf 5566   -1-1->wf1 5567   -onto->wfo 5568   -1-1-onto->wf1o 5569   1stc1st 6771    ~~ cen 7506
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-1st 6773  df-2nd 6774  df-en 7510
This theorem is referenced by:  domssex2  7670  domssex  7671
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