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Theorem domssex2 7732
Description: A corollary of disjenex 7730. 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 5779 . . . . 5  |-  ( F : A -1-1-> B  ->  F : A --> B )
2 fex2 6748 . . . . 5  |-  ( ( F : A --> B  /\  A  e.  V  /\  B  e.  W )  ->  F  e.  _V )
31, 2syl3an1 1301 . . . 4  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  F  e.  _V )
4 f1stres 6815 . . . . . 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 4551 . . . . . . 7  |-  ( B  e.  W  ->  ( B  \  ran  F )  e.  _V )
763ad2ant3 1031 . . . . . 6  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  ( B  \  ran  F )  e. 
_V )
8 snex 4641 . . . . . 6  |-  { ~P U.
ran  A }  e.  _V
9 xpexg 6593 . . . . . 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 668 . . . . 5  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  ( ( B  \  ran  F )  X.  { ~P U. ran  A } )  e. 
_V )
11 fex2 6748 . . . . 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 1268 . . . 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 6592 . . . 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 667 . . 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 6739 . . 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 17 . 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 2451 . . . . . . 7  |-  `' ( F  u.  ( 1st  |`  ( ( B  \  ran  F )  X.  { ~P U. ran  A }
) ) )  =  `' ( F  u.  ( 1st  |`  ( ( B  \  ran  F )  X.  { ~P U. ran  A } ) ) )
1817domss2 7731 . . . . . 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 1020 . . . . 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 5813 . . . . 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 17 . . . 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 3452 . . . 4  |-  ran  `' ( F  u.  ( 1st  |`  ( ( B 
\  ran  F )  X.  { ~P U. ran  A } ) ) ) 
C_  _V
23 f1ss 5784 . . . 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 668 . . 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 1022 . . 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 535 . 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 5774 . . . 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 2453 . . . 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 717 . . 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 3135 . 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 62 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 371    /\ w3a 985    = wceq 1444   E.wex 1663    e. wcel 1887   _Vcvv 3045    \ cdif 3401    u. cun 3402    C_ wss 3404   ~Pcpw 3951   {csn 3968   U.cuni 4198    _I cid 4744    X. cxp 4832   `'ccnv 4833   ran crn 4835    |` cres 4836    o. ccom 4838   -->wf 5578   -1-1->wf1 5579   -1-1-onto->wf1o 5581   1stc1st 6791
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-1st 6793  df-2nd 6794  df-en 7570
This theorem is referenced by: (None)
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