MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  domssex2 Unicode version

Theorem domssex2 7021
Description: A corollary of disjenex 7019. 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 5437 . . . . 5  |-  ( F : A -1-1-> B  ->  F : A --> B )
2 fex2 5401 . . . . 5  |-  ( ( F : A --> B  /\  A  e.  V  /\  B  e.  W )  ->  F  e.  _V )
31, 2syl3an1 1215 . . . 4  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  F  e.  _V )
4 f1stres 6141 . . . . . 6  |-  ( 1st  |`  ( ( B  \  ran  F )  X.  { ~P U. ran  A }
) ) : ( ( B  \  ran  F )  X.  { ~P U.
ran  A } ) --> ( B  \  ran  F )
54a1i 10 . . . . 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 4162 . . . . . . 7  |-  ( B  e.  W  ->  ( B  \  ran  F )  e.  _V )
763ad2ant3 978 . . . . . 6  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  ( B  \  ran  F )  e. 
_V )
8 snex 4216 . . . . . 6  |-  { ~P U.
ran  A }  e.  _V
9 xpexg 4800 . . . . . 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 643 . . . . 5  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  ( ( B  \  ran  F )  X.  { ~P U. ran  A } )  e. 
_V )
11 fex2 5401 . . . . 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 1182 . . . 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 4521 . . . 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 642 . . 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 5208 . . 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 15 . 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 2283 . . . . . . 7  |-  `' ( F  u.  ( 1st  |`  ( ( B  \  ran  F )  X.  { ~P U. ran  A }
) ) )  =  `' ( F  u.  ( 1st  |`  ( ( B  \  ran  F )  X.  { ~P U. ran  A } ) ) )
1817domss2 7020 . . . . . 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 967 . . . . 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 5471 . . . . 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 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-> ran  `' ( F  u.  ( 1st  |`  (
( B  \  ran  F )  X.  { ~P U.
ran  A } ) ) ) )
22 ssv 3198 . . . 4  |-  ran  `' ( F  u.  ( 1st  |`  ( ( B 
\  ran  F )  X.  { ~P U. ran  A } ) ) ) 
C_  _V
23 f1ss 5442 . . . 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 643 . . 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 969 . . 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 518 . 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 5432 . . . 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 4841 . . . . 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 2291 . . . 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 691 . . 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 2869 . 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 56 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 358    /\ w3a 934   E.wex 1528    = wceq 1623    e. wcel 1684   _Vcvv 2788    \ cdif 3149    u. cun 3150    C_ wss 3152   ~Pcpw 3625   {csn 3640   U.cuni 3827    _I cid 4304    X. cxp 4687   `'ccnv 4688   ran crn 4690    |` cres 4691    o. ccom 4693   -->wf 5251   -1-1->wf1 5252   -1-1-onto->wf1o 5254   1stc1st 6120
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
  Copyright terms: Public domain W3C validator