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Theorem domssex2 7226
Description: A corollary of disjenex 7224. 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 5598 . . . . 5  |-  ( F : A -1-1-> B  ->  F : A --> B )
2 fex2 5562 . . . . 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 6327 . . . . . 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 4311 . . . . . . 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 4365 . . . . . 6  |-  { ~P U.
ran  A }  e.  _V
9 xpexg 4948 . . . . . 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 5562 . . . . 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 4669 . . . 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 5364 . . 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 2404 . . . . . . 7  |-  `' ( F  u.  ( 1st  |`  ( ( B  \  ran  F )  X.  { ~P U. ran  A }
) ) )  =  `' ( F  u.  ( 1st  |`  ( ( B  \  ran  F )  X.  { ~P U. ran  A } ) ) )
1817domss2 7225 . . . . . 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 5632 . . . . 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 3328 . . . 4  |-  ran  `' ( F  u.  ( 1st  |`  ( ( B 
\  ran  F )  X.  { ~P U. ran  A } ) ) ) 
C_  _V
23 f1ss 5603 . . . 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 5593 . . . 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 4989 . . . . 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 2412 . . . 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 2997 . 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 1547    = wceq 1649    e. wcel 1721   _Vcvv 2916    \ cdif 3277    u. cun 3278    C_ wss 3280   ~Pcpw 3759   {csn 3774   U.cuni 3975    _I cid 4453    X. cxp 4835   `'ccnv 4836   ran crn 4838    |` cres 4839    o. ccom 4841   -->wf 5409   -1-1->wf1 5410   -1-1-onto->wf1o 5412   1stc1st 6306
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-1st 6308  df-2nd 6309  df-en 7069
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