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Theorem domssex2 7267
Description: A corollary of disjenex 7265. 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 5639 . . . . 5  |-  ( F : A -1-1-> B  ->  F : A --> B )
2 fex2 5603 . . . . 5  |-  ( ( F : A --> B  /\  A  e.  V  /\  B  e.  W )  ->  F  e.  _V )
31, 2syl3an1 1217 . . . 4  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  F  e.  _V )
4 f1stres 6368 . . . . . 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 4351 . . . . . . 7  |-  ( B  e.  W  ->  ( B  \  ran  F )  e.  _V )
763ad2ant3 980 . . . . . 6  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  ( B  \  ran  F )  e. 
_V )
8 snex 4405 . . . . . 6  |-  { ~P U.
ran  A }  e.  _V
9 xpexg 4989 . . . . . 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 644 . . . . 5  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  ( ( B  \  ran  F )  X.  { ~P U. ran  A } )  e. 
_V )
11 fex2 5603 . . . . 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 1184 . . . 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 4710 . . . 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 643 . . 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 5405 . . 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 2436 . . . . . . 7  |-  `' ( F  u.  ( 1st  |`  ( ( B  \  ran  F )  X.  { ~P U. ran  A }
) ) )  =  `' ( F  u.  ( 1st  |`  ( ( B  \  ran  F )  X.  { ~P U. ran  A } ) ) )
1817domss2 7266 . . . . . 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 969 . . . . 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 5673 . . . . 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 3368 . . . 4  |-  ran  `' ( F  u.  ( 1st  |`  ( ( B 
\  ran  F )  X.  { ~P U. ran  A } ) ) ) 
C_  _V
23 f1ss 5644 . . . 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 644 . . 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 971 . . 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 519 . 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 5634 . . . 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 5030 . . . . 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 2444 . . . 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 692 . . 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 3037 . 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 58 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 set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936   E.wex 1550    = wceq 1652    e. wcel 1725   _Vcvv 2956    \ cdif 3317    u. cun 3318    C_ wss 3320   ~Pcpw 3799   {csn 3814   U.cuni 4015    _I cid 4493    X. cxp 4876   `'ccnv 4877   ran crn 4879    |` cres 4880    o. ccom 4882   -->wf 5450   -1-1->wf1 5451   -1-1-onto->wf1o 5453   1stc1st 6347
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-1st 6349  df-2nd 6350  df-en 7110
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