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Theorem mapfien2 27126
Description: Equinumerousity relation for sets of finitely supported functions. MOVABLE (Contributed by Stefan O'Rear, 9-Jul-2015.)
Hypotheses
Ref Expression
mapfien2.s  |-  S  =  { x  e.  ( B  ^m  A )  |  ( `' x " ( _V  \  {  .0.  } ) )  e. 
Fin }
mapfien2.t  |-  T  =  { x  e.  ( D  ^m  C )  |  ( `' x " ( _V  \  { W } ) )  e. 
Fin }
mapfien2.ac  |-  ( ph  ->  A  ~~  C )
mapfien2.bd  |-  ( ph  ->  B  ~~  D )
mapfien2.z  |-  ( ph  ->  .0.  e.  B )
mapfien2.w  |-  ( ph  ->  W  e.  D )
Assertion
Ref Expression
mapfien2  |-  ( ph  ->  S  ~~  T )
Distinct variable groups:    x, A    x, B    x, C    x, D    x,  .0.    x, W
Allowed substitution hints:    ph( x)    S( x)    T( x)

Proof of Theorem mapfien2
Dummy variables  w  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mapfien2.z . . 3  |-  ( ph  ->  .0.  e.  B )
2 mapfien2.w . . 3  |-  ( ph  ->  W  e.  D )
3 mapfien2.bd . . 3  |-  ( ph  ->  B  ~~  D )
4 enfixsn 27125 . . 3  |-  ( (  .0.  e.  B  /\  W  e.  D  /\  B  ~~  D )  ->  E. y ( y : B -1-1-onto-> D  /\  ( y `
 .0.  )  =  W ) )
51, 2, 3, 4syl3anc 1184 . 2  |-  ( ph  ->  E. y ( y : B -1-1-onto-> D  /\  ( y `
 .0.  )  =  W ) )
6 mapfien2.ac . . . . 5  |-  ( ph  ->  A  ~~  C )
7 bren 7076 . . . . 5  |-  ( A 
~~  C  <->  E. z 
z : A -1-1-onto-> C )
86, 7sylib 189 . . . 4  |-  ( ph  ->  E. z  z : A -1-1-onto-> C )
9 mapfien2.s . . . . . . . . . 10  |-  S  =  { x  e.  ( B  ^m  A )  |  ( `' x " ( _V  \  {  .0.  } ) )  e. 
Fin }
10 eqid 2404 . . . . . . . . . 10  |-  { x  e.  ( D  ^m  C
)  |  ( `' x " ( _V 
\  { ( y `
 .0.  ) } ) )  e.  Fin }  =  { x  e.  ( D  ^m  C
)  |  ( `' x " ( _V 
\  { ( y `
 .0.  ) } ) )  e.  Fin }
11 eqid 2404 . . . . . . . . . 10  |-  ( y `
 .0.  )  =  ( y `  .0.  )
12 f1ocnv 5646 . . . . . . . . . . 11  |-  ( z : A -1-1-onto-> C  ->  `' z : C -1-1-onto-> A )
13123ad2ant2 979 . . . . . . . . . 10  |-  ( (
ph  /\  z : A
-1-1-onto-> C  /\  y : B -1-1-onto-> D
)  ->  `' z : C -1-1-onto-> A )
14 simp3 959 . . . . . . . . . 10  |-  ( (
ph  /\  z : A
-1-1-onto-> C  /\  y : B -1-1-onto-> D
)  ->  y : B
-1-1-onto-> D )
1563ad2ant1 978 . . . . . . . . . . 11  |-  ( (
ph  /\  z : A
-1-1-onto-> C  /\  y : B -1-1-onto-> D
)  ->  A  ~~  C )
16 relen 7073 . . . . . . . . . . . 12  |-  Rel  ~~
1716brrelexi 4877 . . . . . . . . . . 11  |-  ( A 
~~  C  ->  A  e.  _V )
1815, 17syl 16 . . . . . . . . . 10  |-  ( (
ph  /\  z : A
-1-1-onto-> C  /\  y : B -1-1-onto-> D
)  ->  A  e.  _V )
1933ad2ant1 978 . . . . . . . . . . 11  |-  ( (
ph  /\  z : A
-1-1-onto-> C  /\  y : B -1-1-onto-> D
)  ->  B  ~~  D )
2016brrelexi 4877 . . . . . . . . . . 11  |-  ( B 
~~  D  ->  B  e.  _V )
2119, 20syl 16 . . . . . . . . . 10  |-  ( (
ph  /\  z : A
-1-1-onto-> C  /\  y : B -1-1-onto-> D
)  ->  B  e.  _V )
2216brrelex2i 4878 . . . . . . . . . . 11  |-  ( A 
~~  C  ->  C  e.  _V )
2315, 22syl 16 . . . . . . . . . 10  |-  ( (
ph  /\  z : A
-1-1-onto-> C  /\  y : B -1-1-onto-> D
)  ->  C  e.  _V )
2416brrelex2i 4878 . . . . . . . . . . 11  |-  ( B 
~~  D  ->  D  e.  _V )
2519, 24syl 16 . . . . . . . . . 10  |-  ( (
ph  /\  z : A
-1-1-onto-> C  /\  y : B -1-1-onto-> D
)  ->  D  e.  _V )
2613ad2ant1 978 . . . . . . . . . 10  |-  ( (
ph  /\  z : A
-1-1-onto-> C  /\  y : B -1-1-onto-> D
)  ->  .0.  e.  B )
279, 10, 11, 13, 14, 18, 21, 23, 25, 26mapfien 7609 . . . . . . . . 9  |-  ( (
ph  /\  z : A
-1-1-onto-> C  /\  y : B -1-1-onto-> D
)  ->  ( w  e.  S  |->  ( y  o.  ( w  o.  `' z ) ) ) : S -1-1-onto-> { x  e.  ( D  ^m  C
)  |  ( `' x " ( _V 
\  { ( y `
 .0.  ) } ) )  e.  Fin } )
28 ovex 6065 . . . . . . . . . . . 12  |-  ( B  ^m  A )  e. 
_V
2928rabex 4314 . . . . . . . . . . 11  |-  { x  e.  ( B  ^m  A
)  |  ( `' x " ( _V 
\  {  .0.  }
) )  e.  Fin }  e.  _V
309, 29eqeltri 2474 . . . . . . . . . 10  |-  S  e. 
_V
3130f1oen 7087 . . . . . . . . 9  |-  ( ( w  e.  S  |->  ( y  o.  ( w  o.  `' z ) ) ) : S -1-1-onto-> {
x  e.  ( D  ^m  C )  |  ( `' x "
( _V  \  {
( y `  .0.  ) } ) )  e. 
Fin }  ->  S  ~~  { x  e.  ( D  ^m  C )  |  ( `' x "
( _V  \  {
( y `  .0.  ) } ) )  e. 
Fin } )
3227, 31syl 16 . . . . . . . 8  |-  ( (
ph  /\  z : A
-1-1-onto-> C  /\  y : B -1-1-onto-> D
)  ->  S  ~~  { x  e.  ( D  ^m  C )  |  ( `' x "
( _V  \  {
( y `  .0.  ) } ) )  e. 
Fin } )
33323adant3r 1181 . . . . . . 7  |-  ( (
ph  /\  z : A
-1-1-onto-> C  /\  ( y : B -1-1-onto-> D  /\  ( y `
 .0.  )  =  W ) )  ->  S  ~~  { x  e.  ( D  ^m  C
)  |  ( `' x " ( _V 
\  { ( y `
 .0.  ) } ) )  e.  Fin } )
34 sneq 3785 . . . . . . . . . . . . . 14  |-  ( ( y `  .0.  )  =  W  ->  { ( y `  .0.  ) }  =  { W } )
3534difeq2d 3425 . . . . . . . . . . . . 13  |-  ( ( y `  .0.  )  =  W  ->  ( _V 
\  { ( y `
 .0.  ) } )  =  ( _V 
\  { W }
) )
3635imaeq2d 5162 . . . . . . . . . . . 12  |-  ( ( y `  .0.  )  =  W  ->  ( `' x " ( _V 
\  { ( y `
 .0.  ) } ) )  =  ( `' x " ( _V 
\  { W }
) ) )
3736eleq1d 2470 . . . . . . . . . . 11  |-  ( ( y `  .0.  )  =  W  ->  ( ( `' x " ( _V 
\  { ( y `
 .0.  ) } ) )  e.  Fin  <->  ( `' x " ( _V 
\  { W }
) )  e.  Fin ) )
3837rabbidv 2908 . . . . . . . . . 10  |-  ( ( y `  .0.  )  =  W  ->  { x  e.  ( D  ^m  C
)  |  ( `' x " ( _V 
\  { ( y `
 .0.  ) } ) )  e.  Fin }  =  { x  e.  ( D  ^m  C
)  |  ( `' x " ( _V 
\  { W }
) )  e.  Fin } )
39 mapfien2.t . . . . . . . . . 10  |-  T  =  { x  e.  ( D  ^m  C )  |  ( `' x " ( _V  \  { W } ) )  e. 
Fin }
4038, 39syl6eqr 2454 . . . . . . . . 9  |-  ( ( y `  .0.  )  =  W  ->  { x  e.  ( D  ^m  C
)  |  ( `' x " ( _V 
\  { ( y `
 .0.  ) } ) )  e.  Fin }  =  T )
4140adantl 453 . . . . . . . 8  |-  ( ( y : B -1-1-onto-> D  /\  ( y `  .0.  )  =  W )  ->  { x  e.  ( D  ^m  C )  |  ( `' x " ( _V  \  {
( y `  .0.  ) } ) )  e. 
Fin }  =  T
)
42413ad2ant3 980 . . . . . . 7  |-  ( (
ph  /\  z : A
-1-1-onto-> C  /\  ( y : B -1-1-onto-> D  /\  ( y `
 .0.  )  =  W ) )  ->  { x  e.  ( D  ^m  C )  |  ( `' x "
( _V  \  {
( y `  .0.  ) } ) )  e. 
Fin }  =  T
)
4333, 42breqtrd 4196 . . . . . 6  |-  ( (
ph  /\  z : A
-1-1-onto-> C  /\  ( y : B -1-1-onto-> D  /\  ( y `
 .0.  )  =  W ) )  ->  S  ~~  T )
44433exp 1152 . . . . 5  |-  ( ph  ->  ( z : A -1-1-onto-> C  ->  ( ( y : B -1-1-onto-> D  /\  ( y `
 .0.  )  =  W )  ->  S  ~~  T ) ) )
4544exlimdv 1643 . . . 4  |-  ( ph  ->  ( E. z  z : A -1-1-onto-> C  ->  ( (
y : B -1-1-onto-> D  /\  ( y `  .0.  )  =  W )  ->  S  ~~  T ) ) )
468, 45mpd 15 . . 3  |-  ( ph  ->  ( ( y : B -1-1-onto-> D  /\  ( y `
 .0.  )  =  W )  ->  S  ~~  T ) )
4746exlimdv 1643 . 2  |-  ( ph  ->  ( E. y ( y : B -1-1-onto-> D  /\  ( y `  .0.  )  =  W )  ->  S  ~~  T ) )
485, 47mpd 15 1  |-  ( ph  ->  S  ~~  T )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936   E.wex 1547    = wceq 1649    e. wcel 1721   {crab 2670   _Vcvv 2916    \ cdif 3277   {csn 3774   class class class wbr 4172    e. cmpt 4226   `'ccnv 4836   "cima 4840    o. ccom 4841   -1-1-onto->wf1o 5412   ` cfv 5413  (class class class)co 6040    ^m cmap 6977    ~~ cen 7065   Fincfn 7068
This theorem is referenced by:  frlmpwfi  27130
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-rep 4280  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-3or 937  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-ral 2671  df-rex 2672  df-reu 2673  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-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  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-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-1o 6683  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-fin 7072
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