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Theorem mapfien2 27226
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 27225 . . 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 7109 . . . . 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 2435 . . . . . . . . . 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 2435 . . . . . . . . . 10  |-  ( y `
 .0.  )  =  ( y `  .0.  )
12 f1ocnv 5679 . . . . . . . . . . 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 7106 . . . . . . . . . . . 12  |-  Rel  ~~
1716brrelexi 4910 . . . . . . . . . . 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 4910 . . . . . . . . . . 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 4911 . . . . . . . . . . 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 4911 . . . . . . . . . . 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 7645 . . . . . . . . 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 6098 . . . . . . . . . . . 12  |-  ( B  ^m  A )  e. 
_V
2928rabex 4346 . . . . . . . . . . 11  |-  { x  e.  ( B  ^m  A
)  |  ( `' x " ( _V 
\  {  .0.  }
) )  e.  Fin }  e.  _V
309, 29eqeltri 2505 . . . . . . . . . 10  |-  S  e. 
_V
3130f1oen 7120 . . . . . . . . 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 3817 . . . . . . . . . . . . . 14  |-  ( ( y `  .0.  )  =  W  ->  { ( y `  .0.  ) }  =  { W } )
3534difeq2d 3457 . . . . . . . . . . . . 13  |-  ( ( y `  .0.  )  =  W  ->  ( _V 
\  { ( y `
 .0.  ) } )  =  ( _V 
\  { W }
) )
3635imaeq2d 5195 . . . . . . . . . . . 12  |-  ( ( y `  .0.  )  =  W  ->  ( `' x " ( _V 
\  { ( y `
 .0.  ) } ) )  =  ( `' x " ( _V 
\  { W }
) ) )
3736eleq1d 2501 . . . . . . . . . . 11  |-  ( ( y `  .0.  )  =  W  ->  ( ( `' x " ( _V 
\  { ( y `
 .0.  ) } ) )  e.  Fin  <->  ( `' x " ( _V 
\  { W }
) )  e.  Fin ) )
3837rabbidv 2940 . . . . . . . . . 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 2485 . . . . . . . . 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 4228 . . . . . 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 1646 . . . 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 1646 . 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 1550    = wceq 1652    e. wcel 1725   {crab 2701   _Vcvv 2948    \ cdif 3309   {csn 3806   class class class wbr 4204    e. cmpt 4258   `'ccnv 4869   "cima 4873    o. ccom 4874   -1-1-onto->wf1o 5445   ` cfv 5446  (class class class)co 6073    ^m cmap 7010    ~~ cen 7098   Fincfn 7101
This theorem is referenced by:  frlmpwfi  27230
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 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-1o 6716  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-fin 7105
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