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Theorem domss2 7268
Description: A corollary of disjenex 7267. 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 5687 . . . . . . . 8  |-  ( F : A -1-1-> B  ->  F : A -1-1-onto-> ran  F )
213ad2ant1 979 . . . . . . 7  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  F : A
-1-1-onto-> ran  F )
3 simp2 959 . . . . . . . . . 10  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  A  e.  V )
4 rnexg 5133 . . . . . . . . . 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 4708 . . . . . . . . 9  |-  ( ran 
A  e.  _V  ->  U.
ran  A  e.  _V )
7 pwexg 4385 . . . . . . . . 9  |-  ( U. ran  A  e.  _V  ->  ~P
U. ran  A  e.  _V )
85, 6, 73syl 19 . . . . . . . 8  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  ~P U. ran  A  e.  _V )
9 1stconst 6437 . . . . . . . 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 4353 . . . . . . . . . 10  |-  ( B  e.  W  ->  ( B  \  ran  F )  e.  _V )
12113ad2ant3 981 . . . . . . . . 9  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  ( B  \  ran  F )  e. 
_V )
13 disjen 7266 . . . . . . . . 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 644 . . . . . . . 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 447 . . . . . . 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 3702 . . . . . . . 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 5696 . . . . . . 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 1186 . . . . . 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 3706 . . . . . . . 8  |-  ( ran 
F  u.  ( B 
\  ran  F )
)  =  ( ran 
F  u.  B )
21 f1f 5641 . . . . . . . . . . 11  |-  ( F : A -1-1-> B  ->  F : A --> B )
22213ad2ant1 979 . . . . . . . . . 10  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  F : A
--> B )
23 frn 5599 . . . . . . . . . 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 3519 . . . . . . . . 9  |-  ( ran 
F  C_  B  <->  ( ran  F  u.  B )  =  B )
2624, 25sylib 190 . . . . . . . 8  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  ( ran  F  u.  B )  =  B )
2720, 26syl5eq 2482 . . . . . . 7  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  ( ran  F  u.  ( B  \  ran  F ) )  =  B )
28 f1oeq3 5669 . . . . . . 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 203 . . . . 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 5689 . . . . 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 5667 . . . . 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 8 . . . 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 205 . . 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 5683 . . . . 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 5658 . . . . 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 19 . . . 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 5669 . . . 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 225 . 2  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  G : B
-1-1-onto-> ran  G )
43 ssun1 3512 . . 3  |-  A  C_  ( A  u.  (
( B  \  ran  F )  X.  { ~P U.
ran  A } ) )
4443, 39syl5sseqr 3399 . 2  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  A  C_  ran  G )
45 ssid 3369 . . . 4  |-  ran  F  C_ 
ran  F
46 cores 5375 . . . 4  |-  ( ran 
F  C_  ran  F  -> 
( ( G  |`  ran  F )  o.  F
)  =  ( G  o.  F ) )
4745, 46ax-mp 8 . . 3  |-  ( ( G  |`  ran  F )  o.  F )  =  ( G  o.  F
)
48 dmres 5169 . . . . . . . . 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 5689 . . . . . . . . . . . 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 5680 . . . . . . . . . . . 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 19 . . . . . . . . . . 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 3544 . . . . . . . . . 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 2486 . . . . . . . . 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 2482 . . . . . . . 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 5176 . . . . . . . . 9  |-  Rel  ( `' ( 1st  |`  (
( B  \  ran  F )  X.  { ~P U.
ran  A } ) )  |`  ran  F )
56 reldm0 5089 . . . . . . . . 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 8 . . . . . . . 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 205 . . . . . . 7  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  ( `' ( 1st  |`  ( ( B  \  ran  F )  X.  { ~P U. ran  A } ) )  |`  ran  F )  =  (/) )
5958uneq2d 3503 . . . . . 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 5279 . . . . . . . . 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 2458 . . . . . . . 8  |-  G  =  ( `' F  u.  `' ( 1st  |`  (
( B  \  ran  F )  X.  { ~P U.
ran  A } ) ) )
6261reseq1i 5144 . . . . . . 7  |-  ( G  |`  ran  F )  =  ( ( `' F  u.  `' ( 1st  |`  (
( B  \  ran  F )  X.  { ~P U.
ran  A } ) ) )  |`  ran  F
)
63 resundir 5163 . . . . . . 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 4891 . . . . . . . . . 10  |-  ran  F  =  dom  `' F
6564reseq2i 5145 . . . . . . . . 9  |-  ( `' F  |`  ran  F )  =  ( `' F  |` 
dom  `' F )
66 relcnv 5244 . . . . . . . . . 10  |-  Rel  `' F
67 resdm 5186 . . . . . . . . . 10  |-  ( Rel  `' F  ->  ( `' F  |`  dom  `' F
)  =  `' F
)
6866, 67ax-mp 8 . . . . . . . . 9  |-  ( `' F  |`  dom  `' F
)  =  `' F
6965, 68eqtri 2458 . . . . . . . 8  |-  ( `' F  |`  ran  F )  =  `' F
7069uneq1i 3499 . . . . . . 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 2463 . . . . . 6  |-  ( `' F  u.  ( `' ( 1st  |`  (
( B  \  ran  F )  X.  { ~P U.
ran  A } ) )  |`  ran  F ) )  =  ( G  |`  ran  F )
72 un0 3654 . . . . . 6  |-  ( `' F  u.  (/) )  =  `' F
7359, 71, 723eqtr3g 2493 . . . . 5  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  ( G  |` 
ran  F )  =  `' F )
7473coeq1d 5036 . . . 4  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  ( ( G  |`  ran  F )  o.  F )  =  ( `' F  o.  F ) )
75 f1cocnv1 5707 . . . . 5  |-  ( F : A -1-1-> B  -> 
( `' F  o.  F )  =  (  _I  |`  A )
)
76753ad2ant1 979 . . . 4  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  ( `' F  o.  F )  =  (  _I  |`  A ) )
7774, 76eqtrd 2470 . . 3  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  ( ( G  |`  ran  F )  o.  F )  =  (  _I  |`  A ) )
7847, 77syl5eqr 2484 . 2  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  ( G  o.  F )  =  (  _I  |`  A )
)
7942, 44, 783jca 1135 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 set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   _Vcvv 2958    \ cdif 3319    u. cun 3320    i^i cin 3321    C_ wss 3322   (/)c0 3630   ~Pcpw 3801   {csn 3816   U.cuni 4017   class class class wbr 4214    _I cid 4495    X. cxp 4878   `'ccnv 4879   dom cdm 4880   ran crn 4881    |` cres 4882    o. ccom 4884   Rel wrel 4885   -->wf 5452   -1-1->wf1 5453   -onto->wfo 5454   -1-1-onto->wf1o 5455   1stc1st 6349    ~~ cen 7108
This theorem is referenced by:  domssex2  7269  domssex  7270
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-1st 6351  df-2nd 6352  df-en 7112
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