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Theorem domssex2 7463
Description: A corollary of disjenex 7461. If  F is an injection from  A to  B then there is a right inverse  g of  F from  B to a superset of  A. (Contributed by Mario Carneiro, 7-Feb-2015.) (Revised by Mario Carneiro, 24-Jun-2015.)
Assertion
Ref Expression
domssex2  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  E. g
( g : B -1-1-> _V 
/\  ( g  o.  F )  =  (  _I  |`  A )
) )
Distinct variable groups:    A, g    B, g    g, F
Allowed substitution hints:    V( g)    W( g)

Proof of Theorem domssex2
StepHypRef Expression
1 f1f 5601 . . . . 5  |-  ( F : A -1-1-> B  ->  F : A --> B )
2 fex2 6527 . . . . 5  |-  ( ( F : A --> B  /\  A  e.  V  /\  B  e.  W )  ->  F  e.  _V )
31, 2syl3an1 1251 . . . 4  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  F  e.  _V )
4 f1stres 6593 . . . . . 6  |-  ( 1st  |`  ( ( B  \  ran  F )  X.  { ~P U. ran  A }
) ) : ( ( B  \  ran  F )  X.  { ~P U.
ran  A } ) --> ( B  \  ran  F )
54a1i 11 . . . . 5  |-  ( ( 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 } ) --> ( B  \  ran  F ) )
6 difexg 4435 . . . . . . 7  |-  ( B  e.  W  ->  ( B  \  ran  F )  e.  _V )
763ad2ant3 1011 . . . . . 6  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  ( B  \  ran  F )  e. 
_V )
8 snex 4528 . . . . . 6  |-  { ~P U.
ran  A }  e.  _V
9 xpexg 6502 . . . . . 6  |-  ( ( ( B  \  ran  F )  e.  _V  /\  { ~P U. ran  A }  e.  _V )  ->  ( ( B  \  ran  F )  X.  { ~P U. ran  A }
)  e.  _V )
107, 8, 9sylancl 662 . . . . 5  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  ( ( B  \  ran  F )  X.  { ~P U. ran  A } )  e. 
_V )
11 fex2 6527 . . . . 5  |-  ( ( ( 1st  |`  (
( B  \  ran  F )  X.  { ~P U.
ran  A } ) ) : ( ( B  \  ran  F
)  X.  { ~P U.
ran  A } ) --> ( B  \  ran  F )  /\  ( ( B  \  ran  F
)  X.  { ~P U.
ran  A } )  e.  _V  /\  ( B  \  ran  F )  e.  _V )  -> 
( 1st  |`  ( ( B  \  ran  F
)  X.  { ~P U.
ran  A } ) )  e.  _V )
125, 10, 7, 11syl3anc 1218 . . . 4  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  ( 1st  |`  ( ( B  \  ran  F )  X.  { ~P U. ran  A }
) )  e.  _V )
13 unexg 6376 . . . 4  |-  ( ( F  e.  _V  /\  ( 1st  |`  ( ( B  \  ran  F )  X.  { ~P U. ran  A } ) )  e.  _V )  -> 
( F  u.  ( 1st  |`  ( ( B 
\  ran  F )  X.  { ~P U. ran  A } ) ) )  e.  _V )
143, 12, 13syl2anc 661 . . 3  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  ( F  u.  ( 1st  |`  (
( B  \  ran  F )  X.  { ~P U.
ran  A } ) ) )  e.  _V )
15 cnvexg 6519 . . 3  |-  ( ( F  u.  ( 1st  |`  ( ( B  \  ran  F )  X.  { ~P U. ran  A }
) ) )  e. 
_V  ->  `' ( F  u.  ( 1st  |`  (
( B  \  ran  F )  X.  { ~P U.
ran  A } ) ) )  e.  _V )
1614, 15syl 16 . 2  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  `' ( F  u.  ( 1st  |`  ( ( B  \  ran  F )  X.  { ~P U. ran  A }
) ) )  e. 
_V )
17 eqid 2438 . . . . . . 7  |-  `' ( F  u.  ( 1st  |`  ( ( B  \  ran  F )  X.  { ~P U. ran  A }
) ) )  =  `' ( F  u.  ( 1st  |`  ( ( B  \  ran  F )  X.  { ~P U. ran  A } ) ) )
1817domss2 7462 . . . . . 6  |-  ( ( 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-> ran  `' ( F  u.  ( 1st  |`  (
( B  \  ran  F )  X.  { ~P U.
ran  A } ) ) )  /\  A  C_ 
ran  `' ( F  u.  ( 1st  |`  ( ( B  \  ran  F )  X.  { ~P U. ran  A } ) ) )  /\  ( `' ( F  u.  ( 1st  |`  ( ( B 
\  ran  F )  X.  { ~P U. ran  A } ) ) )  o.  F )  =  (  _I  |`  A ) ) )
1918simp1d 1000 . . . . 5  |-  ( ( 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-> ran  `' ( F  u.  ( 1st  |`  (
( B  \  ran  F )  X.  { ~P U.
ran  A } ) ) ) )
20 f1of1 5635 . . . . 5  |-  ( `' ( F  u.  ( 1st  |`  ( ( B 
\  ran  F )  X.  { ~P U. ran  A } ) ) ) : B -1-1-onto-> ran  `' ( F  u.  ( 1st  |`  (
( B  \  ran  F )  X.  { ~P U.
ran  A } ) ) )  ->  `' ( F  u.  ( 1st  |`  ( ( B 
\  ran  F )  X.  { ~P U. ran  A } ) ) ) : B -1-1-> ran  `' ( F  u.  ( 1st  |`  ( ( B 
\  ran  F )  X.  { ~P U. ran  A } ) ) ) )
2119, 20syl 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-> ran  `' ( F  u.  ( 1st  |`  (
( B  \  ran  F )  X.  { ~P U.
ran  A } ) ) ) )
22 ssv 3371 . . . 4  |-  ran  `' ( F  u.  ( 1st  |`  ( ( B 
\  ran  F )  X.  { ~P U. ran  A } ) ) ) 
C_  _V
23 f1ss 5606 . . . 4  |-  ( ( `' ( F  u.  ( 1st  |`  ( ( B  \  ran  F )  X.  { ~P U. ran  A } ) ) ) : B -1-1-> ran  `' ( F  u.  ( 1st  |`  ( ( B 
\  ran  F )  X.  { ~P U. ran  A } ) ) )  /\  ran  `' ( F  u.  ( 1st  |`  ( ( B  \  ran  F )  X.  { ~P U. ran  A }
) ) )  C_  _V )  ->  `' ( F  u.  ( 1st  |`  ( ( B  \  ran  F )  X.  { ~P U. ran  A }
) ) ) : B -1-1-> _V )
2421, 22, 23sylancl 662 . . 3  |-  ( ( 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-> _V )
2518simp3d 1002 . . 3  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  ( `' ( F  u.  ( 1st  |`  ( ( B 
\  ran  F )  X.  { ~P U. ran  A } ) ) )  o.  F )  =  (  _I  |`  A ) )
2624, 25jca 532 . 2  |-  ( ( 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-> _V  /\  ( `' ( F  u.  ( 1st  |`  ( ( B  \  ran  F )  X.  { ~P U. ran  A } ) ) )  o.  F )  =  (  _I  |`  A ) ) )
27 f1eq1 5596 . . . 4  |-  ( g  =  `' ( F  u.  ( 1st  |`  (
( B  \  ran  F )  X.  { ~P U.
ran  A } ) ) )  ->  (
g : B -1-1-> _V  <->  `' ( F  u.  ( 1st  |`  ( ( B 
\  ran  F )  X.  { ~P U. ran  A } ) ) ) : B -1-1-> _V )
)
28 coeq1 4992 . . . . 5  |-  ( g  =  `' ( F  u.  ( 1st  |`  (
( B  \  ran  F )  X.  { ~P U.
ran  A } ) ) )  ->  (
g  o.  F )  =  ( `' ( F  u.  ( 1st  |`  ( ( B  \  ran  F )  X.  { ~P U. ran  A }
) ) )  o.  F ) )
2928eqeq1d 2446 . . . 4  |-  ( g  =  `' ( F  u.  ( 1st  |`  (
( B  \  ran  F )  X.  { ~P U.
ran  A } ) ) )  ->  (
( g  o.  F
)  =  (  _I  |`  A )  <->  ( `' ( F  u.  ( 1st  |`  ( ( B 
\  ran  F )  X.  { ~P U. ran  A } ) ) )  o.  F )  =  (  _I  |`  A ) ) )
3027, 29anbi12d 710 . . 3  |-  ( g  =  `' ( F  u.  ( 1st  |`  (
( B  \  ran  F )  X.  { ~P U.
ran  A } ) ) )  ->  (
( g : B -1-1-> _V 
/\  ( g  o.  F )  =  (  _I  |`  A )
)  <->  ( `' ( F  u.  ( 1st  |`  ( ( B  \  ran  F )  X.  { ~P U. ran  A }
) ) ) : B -1-1-> _V  /\  ( `' ( F  u.  ( 1st  |`  ( ( B  \  ran  F )  X.  { ~P U. ran  A } ) ) )  o.  F )  =  (  _I  |`  A ) ) ) )
3130spcegv 3053 . 2  |-  ( `' ( F  u.  ( 1st  |`  ( ( B 
\  ran  F )  X.  { ~P U. ran  A } ) ) )  e.  _V  ->  (
( `' ( F  u.  ( 1st  |`  (
( B  \  ran  F )  X.  { ~P U.
ran  A } ) ) ) : B -1-1-> _V 
/\  ( `' ( F  u.  ( 1st  |`  ( ( B  \  ran  F )  X.  { ~P U. ran  A }
) ) )  o.  F )  =  (  _I  |`  A )
)  ->  E. g
( g : B -1-1-> _V 
/\  ( g  o.  F )  =  (  _I  |`  A )
) ) )
3216, 26, 31sylc 60 1  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  E. g
( g : B -1-1-> _V 
/\  ( g  o.  F )  =  (  _I  |`  A )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369   E.wex 1586    e. wcel 1756   _Vcvv 2967    \ cdif 3320    u. cun 3321    C_ wss 3323   ~Pcpw 3855   {csn 3872   U.cuni 4086    _I cid 4626    X. cxp 4833   `'ccnv 4834   ran crn 4836    |` cres 4837    o. ccom 4839   -->wf 5409   -1-1->wf1 5410   -1-1-onto->wf1o 5412   1stc1st 6570
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-1st 6572  df-2nd 6573  df-en 7303
This theorem is referenced by: (None)
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