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Theorem domss2 7020
Description: A corollary of disjenex 7019. 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 5483 . . . . . . . 8  |-  ( F : A -1-1-> B  ->  F : A -1-1-onto-> ran  F )
213ad2ant1 976 . . . . . . 7  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  F : A
-1-1-onto-> ran  F )
3 simp2 956 . . . . . . . . . 10  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  A  e.  V )
4 rnexg 4940 . . . . . . . . . 10  |-  ( A  e.  V  ->  ran  A  e.  _V )
53, 4syl 15 . . . . . . . . 9  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  ran  A  e. 
_V )
6 uniexg 4517 . . . . . . . . 9  |-  ( ran 
A  e.  _V  ->  U.
ran  A  e.  _V )
7 pwexg 4194 . . . . . . . . 9  |-  ( U. ran  A  e.  _V  ->  ~P
U. ran  A  e.  _V )
85, 6, 73syl 18 . . . . . . . 8  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  ~P U. ran  A  e.  _V )
9 1stconst 6207 . . . . . . . 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 15 . . . . . . 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 4162 . . . . . . . . . 10  |-  ( B  e.  W  ->  ( B  \  ran  F )  e.  _V )
12113ad2ant3 978 . . . . . . . . 9  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  ( B  \  ran  F )  e. 
_V )
13 disjen 7018 . . . . . . . . 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 642 . . . . . . . 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 445 . . . . . . 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 3526 . . . . . . . 8  |-  ( ran 
F  i^i  ( B  \  ran  F ) )  =  (/)
1716a1i 10 . . . . . . 7  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  ( ran  F  i^i  ( B  \  ran  F ) )  =  (/) )
18 f1oun 5492 . . . . . . 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 1183 . . . . . 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 3530 . . . . . . . 8  |-  ( ran 
F  u.  ( B 
\  ran  F )
)  =  ( ran 
F  u.  B )
21 f1f 5437 . . . . . . . . . . 11  |-  ( F : A -1-1-> B  ->  F : A --> B )
22213ad2ant1 976 . . . . . . . . . 10  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  F : A
--> B )
23 frn 5395 . . . . . . . . . 10  |-  ( F : A --> B  ->  ran  F  C_  B )
2422, 23syl 15 . . . . . . . . 9  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  ran  F  C_  B )
25 ssequn1 3345 . . . . . . . . 9  |-  ( ran 
F  C_  B  <->  ( ran  F  u.  B )  =  B )
2624, 25sylib 188 . . . . . . . 8  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  ( ran  F  u.  B )  =  B )
2720, 26syl5eq 2327 . . . . . . 7  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  ( ran  F  u.  ( B  \  ran  F ) )  =  B )
28 f1oeq3 5465 . . . . . . 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 15 . . . . . 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 201 . . . . 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 5485 . . . . 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 15 . . . 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 5463 . . . . 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 203 . . 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 5479 . . . . 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 5454 . . . . 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 18 . . . 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 5465 . . . 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 15 . . 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 223 . 2  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  G : B
-1-1-onto-> ran  G )
43 ssun1 3338 . . 3  |-  A  C_  ( A  u.  (
( B  \  ran  F )  X.  { ~P U.
ran  A } ) )
4443, 39syl5sseqr 3227 . 2  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  A  C_  ran  G )
45 ssid 3197 . . . 4  |-  ran  F  C_ 
ran  F
46 cores 5176 . . . 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 4976 . . . . . . . . 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 5485 . . . . . . . . . . . 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 5476 . . . . . . . . . . . 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 18 . . . . . . . . . . 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 3370 . . . . . . . . . 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 2331 . . . . . . . . 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 2327 . . . . . . . 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 4983 . . . . . . . . 9  |-  Rel  ( `' ( 1st  |`  (
( B  \  ran  F )  X.  { ~P U.
ran  A } ) )  |`  ran  F )
56 reldm0 4896 . . . . . . . . 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 203 . . . . . . 7  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  ( `' ( 1st  |`  ( ( B  \  ran  F )  X.  { ~P U. ran  A } ) )  |`  ran  F )  =  (/) )
5958uneq2d 3329 . . . . . 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 5086 . . . . . . . . 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 2303 . . . . . . . 8  |-  G  =  ( `' F  u.  `' ( 1st  |`  (
( B  \  ran  F )  X.  { ~P U.
ran  A } ) ) )
6261reseq1i 4951 . . . . . . 7  |-  ( G  |`  ran  F )  =  ( ( `' F  u.  `' ( 1st  |`  (
( B  \  ran  F )  X.  { ~P U.
ran  A } ) ) )  |`  ran  F
)
63 resundir 4970 . . . . . . 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 4700 . . . . . . . . . 10  |-  ran  F  =  dom  `' F
6564reseq2i 4952 . . . . . . . . 9  |-  ( `' F  |`  ran  F )  =  ( `' F  |` 
dom  `' F )
66 relcnv 5051 . . . . . . . . . 10  |-  Rel  `' F
67 resdm 4993 . . . . . . . . . 10  |-  ( Rel  `' F  ->  ( `' F  |`  dom  `' F
)  =  `' F
)
6866, 67ax-mp 8 . . . . . . . . 9  |-  ( `' F  |`  dom  `' F
)  =  `' F
6965, 68eqtri 2303 . . . . . . . 8  |-  ( `' F  |`  ran  F )  =  `' F
7069uneq1i 3325 . . . . . . 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 2308 . . . . . 6  |-  ( `' F  u.  ( `' ( 1st  |`  (
( B  \  ran  F )  X.  { ~P U.
ran  A } ) )  |`  ran  F ) )  =  ( G  |`  ran  F )
72 un0 3479 . . . . . 6  |-  ( `' F  u.  (/) )  =  `' F
7359, 71, 723eqtr3g 2338 . . . . 5  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  ( G  |` 
ran  F )  =  `' F )
7473coeq1d 4845 . . . 4  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  ( ( G  |`  ran  F )  o.  F )  =  ( `' F  o.  F ) )
75 f1cocnv1 5503 . . . . 5  |-  ( F : A -1-1-> B  -> 
( `' F  o.  F )  =  (  _I  |`  A )
)
76753ad2ant1 976 . . . 4  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  ( `' F  o.  F )  =  (  _I  |`  A ) )
7774, 76eqtrd 2315 . . 3  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  ( ( G  |`  ran  F )  o.  F )  =  (  _I  |`  A ) )
7847, 77syl5eqr 2329 . 2  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  ( G  o.  F )  =  (  _I  |`  A )
)
7942, 44, 783jca 1132 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   _Vcvv 2788    \ cdif 3149    u. cun 3150    i^i cin 3151    C_ wss 3152   (/)c0 3455   ~Pcpw 3625   {csn 3640   U.cuni 3827   class class class wbr 4023    _I cid 4304    X. cxp 4687   `'ccnv 4688   dom cdm 4689   ran crn 4690    |` cres 4691    o. ccom 4693   Rel wrel 4694   -->wf 5251   -1-1->wf1 5252   -onto->wfo 5253   -1-1-onto->wf1o 5254   1stc1st 6120    ~~ cen 6860
This theorem is referenced by:  domssex2  7021  domssex  7022
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-1st 6122  df-2nd 6123  df-en 6864
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