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Theorem mapfien2 7650
Description: Equinumerousity relation for sets of finitely supported functions. (Contributed by Stefan O'Rear, 9-Jul-2015.) (Revised by AV, 7-Jul-2019.)
Hypotheses
Ref Expression
mapfien2.s  |-  S  =  { x  e.  ( B  ^m  A )  |  x finSupp  .0.  }
mapfien2.t  |-  T  =  { x  e.  ( D  ^m  C )  |  x finSupp  W }
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 7412 . . 3  |-  ( (  .0.  e.  B  /\  W  e.  D  /\  B  ~~  D )  ->  E. y ( y : B -1-1-onto-> D  /\  ( y `
 .0.  )  =  W ) )
51, 2, 3, 4syl3anc 1218 . 2  |-  ( ph  ->  E. y ( y : B -1-1-onto-> D  /\  ( y `
 .0.  )  =  W ) )
6 mapfien2.ac . . . . 5  |-  ( ph  ->  A  ~~  C )
7 bren 7311 . . . . 5  |-  ( A 
~~  C  <->  E. z 
z : A -1-1-onto-> C )
86, 7sylib 196 . . . 4  |-  ( ph  ->  E. z  z : A -1-1-onto-> C )
9 mapfien2.s . . . . . . . . . 10  |-  S  =  { x  e.  ( B  ^m  A )  |  x finSupp  .0.  }
10 eqid 2438 . . . . . . . . . 10  |-  { x  e.  ( D  ^m  C
)  |  x finSupp  (
y `  .0.  ) }  =  { x  e.  ( D  ^m  C
)  |  x finSupp  (
y `  .0.  ) }
11 eqid 2438 . . . . . . . . . 10  |-  ( y `
 .0.  )  =  ( y `  .0.  )
12 f1ocnv 5648 . . . . . . . . . . 11  |-  ( z : A -1-1-onto-> C  ->  `' z : C -1-1-onto-> A )
13123ad2ant2 1010 . . . . . . . . . 10  |-  ( (
ph  /\  z : A
-1-1-onto-> C  /\  y : B -1-1-onto-> D
)  ->  `' z : C -1-1-onto-> A )
14 simp3 990 . . . . . . . . . 10  |-  ( (
ph  /\  z : A
-1-1-onto-> C  /\  y : B -1-1-onto-> D
)  ->  y : B
-1-1-onto-> D )
1563ad2ant1 1009 . . . . . . . . . . 11  |-  ( (
ph  /\  z : A
-1-1-onto-> C  /\  y : B -1-1-onto-> D
)  ->  A  ~~  C )
16 relen 7307 . . . . . . . . . . . 12  |-  Rel  ~~
1716brrelexi 4874 . . . . . . . . . . 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 1009 . . . . . . . . . . 11  |-  ( (
ph  /\  z : A
-1-1-onto-> C  /\  y : B -1-1-onto-> D
)  ->  B  ~~  D )
2016brrelexi 4874 . . . . . . . . . . 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 4875 . . . . . . . . . . 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 4875 . . . . . . . . . . 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 1009 . . . . . . . . . 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 7649 . . . . . . . . 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 finSupp  (
y `  .0.  ) } )
28 ovex 6111 . . . . . . . . . . . 12  |-  ( B  ^m  A )  e. 
_V
2928rabex 4438 . . . . . . . . . . 11  |-  { x  e.  ( B  ^m  A
)  |  x finSupp  .0.  }  e.  _V
309, 29eqeltri 2508 . . . . . . . . . 10  |-  S  e. 
_V
3130f1oen 7322 . . . . . . . . 9  |-  ( ( w  e.  S  |->  ( y  o.  ( w  o.  `' z ) ) ) : S -1-1-onto-> {
x  e.  ( D  ^m  C )  |  x finSupp  ( y `  .0.  ) }  ->  S  ~~  { x  e.  ( D  ^m  C )  |  x finSupp  ( y `  .0.  ) } )
3227, 31syl 16 . . . . . . . 8  |-  ( (
ph  /\  z : A
-1-1-onto-> C  /\  y : B -1-1-onto-> D
)  ->  S  ~~  { x  e.  ( D  ^m  C )  |  x finSupp  ( y `  .0.  ) } )
33323adant3r 1215 . . . . . . 7  |-  ( (
ph  /\  z : A
-1-1-onto-> C  /\  ( y : B -1-1-onto-> D  /\  ( y `
 .0.  )  =  W ) )  ->  S  ~~  { x  e.  ( D  ^m  C
)  |  x finSupp  (
y `  .0.  ) } )
34 breq2 4291 . . . . . . . . . . 11  |-  ( ( y `  .0.  )  =  W  ->  ( x finSupp 
( y `  .0.  ) 
<->  x finSupp  W ) )
3534rabbidv 2959 . . . . . . . . . 10  |-  ( ( y `  .0.  )  =  W  ->  { x  e.  ( D  ^m  C
)  |  x finSupp  (
y `  .0.  ) }  =  { x  e.  ( D  ^m  C
)  |  x finSupp  W } )
36 mapfien2.t . . . . . . . . . 10  |-  T  =  { x  e.  ( D  ^m  C )  |  x finSupp  W }
3735, 36syl6eqr 2488 . . . . . . . . 9  |-  ( ( y `  .0.  )  =  W  ->  { x  e.  ( D  ^m  C
)  |  x finSupp  (
y `  .0.  ) }  =  T )
3837adantl 466 . . . . . . . 8  |-  ( ( y : B -1-1-onto-> D  /\  ( y `  .0.  )  =  W )  ->  { x  e.  ( D  ^m  C )  |  x finSupp  ( y `  .0.  ) }  =  T )
39383ad2ant3 1011 . . . . . . 7  |-  ( (
ph  /\  z : A
-1-1-onto-> C  /\  ( y : B -1-1-onto-> D  /\  ( y `
 .0.  )  =  W ) )  ->  { x  e.  ( D  ^m  C )  |  x finSupp  ( y `  .0.  ) }  =  T )
4033, 39breqtrd 4311 . . . . . 6  |-  ( (
ph  /\  z : A
-1-1-onto-> C  /\  ( y : B -1-1-onto-> D  /\  ( y `
 .0.  )  =  W ) )  ->  S  ~~  T )
41403exp 1186 . . . . 5  |-  ( ph  ->  ( z : A -1-1-onto-> C  ->  ( ( y : B -1-1-onto-> D  /\  ( y `
 .0.  )  =  W )  ->  S  ~~  T ) ) )
4241exlimdv 1690 . . . 4  |-  ( ph  ->  ( E. z  z : A -1-1-onto-> C  ->  ( (
y : B -1-1-onto-> D  /\  ( y `  .0.  )  =  W )  ->  S  ~~  T ) ) )
438, 42mpd 15 . . 3  |-  ( ph  ->  ( ( y : B -1-1-onto-> D  /\  ( y `
 .0.  )  =  W )  ->  S  ~~  T ) )
4443exlimdv 1690 . 2  |-  ( ph  ->  ( E. y ( y : B -1-1-onto-> D  /\  ( y `  .0.  )  =  W )  ->  S  ~~  T ) )
455, 44mpd 15 1  |-  ( ph  ->  S  ~~  T )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369   E.wex 1586    e. wcel 1756   {crab 2714   _Vcvv 2967   class class class wbr 4287    e. cmpt 4345   `'ccnv 4834    o. ccom 4839   -1-1-onto->wf1o 5412   ` cfv 5413  (class class class)co 6086    ^m cmap 7206    ~~ cen 7299   finSupp cfsupp 7612
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-supp 6686  df-1o 6912  df-er 7093  df-map 7208  df-en 7303  df-dom 7304  df-fin 7306  df-fsupp 7613
This theorem is referenced by: (None)
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