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Theorem mapfien2 27361
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 27360 . . 3  |-  ( (  .0.  e.  B  /\  W  e.  D  /\  B  ~~  D )  ->  E. y ( y : B -1-1-onto-> D  /\  ( y `
 .0.  )  =  W ) )
51, 2, 3, 4syl3anc 1182 . 2  |-  ( ph  ->  E. y ( y : B -1-1-onto-> D  /\  ( y `
 .0.  )  =  W ) )
6 mapfien2.ac . . . . 5  |-  ( ph  ->  A  ~~  C )
7 bren 6887 . . . . 5  |-  ( A 
~~  C  <->  E. z 
z : A -1-1-onto-> C )
86, 7sylib 188 . . . 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 2296 . . . . . . . . . 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 2296 . . . . . . . . . 10  |-  ( y `
 .0.  )  =  ( y `  .0.  )
12 f1ocnv 5501 . . . . . . . . . . 11  |-  ( z : A -1-1-onto-> C  ->  `' z : C -1-1-onto-> A )
13123ad2ant2 977 . . . . . . . . . 10  |-  ( (
ph  /\  z : A
-1-1-onto-> C  /\  y : B -1-1-onto-> D
)  ->  `' z : C -1-1-onto-> A )
14 simp3 957 . . . . . . . . . 10  |-  ( (
ph  /\  z : A
-1-1-onto-> C  /\  y : B -1-1-onto-> D
)  ->  y : B
-1-1-onto-> D )
1563ad2ant1 976 . . . . . . . . . . 11  |-  ( (
ph  /\  z : A
-1-1-onto-> C  /\  y : B -1-1-onto-> D
)  ->  A  ~~  C )
16 relen 6884 . . . . . . . . . . . 12  |-  Rel  ~~
1716brrelexi 4745 . . . . . . . . . . 11  |-  ( A 
~~  C  ->  A  e.  _V )
1815, 17syl 15 . . . . . . . . . 10  |-  ( (
ph  /\  z : A
-1-1-onto-> C  /\  y : B -1-1-onto-> D
)  ->  A  e.  _V )
1933ad2ant1 976 . . . . . . . . . . 11  |-  ( (
ph  /\  z : A
-1-1-onto-> C  /\  y : B -1-1-onto-> D
)  ->  B  ~~  D )
2016brrelexi 4745 . . . . . . . . . . 11  |-  ( B 
~~  D  ->  B  e.  _V )
2119, 20syl 15 . . . . . . . . . 10  |-  ( (
ph  /\  z : A
-1-1-onto-> C  /\  y : B -1-1-onto-> D
)  ->  B  e.  _V )
2216brrelex2i 4746 . . . . . . . . . . 11  |-  ( A 
~~  C  ->  C  e.  _V )
2315, 22syl 15 . . . . . . . . . 10  |-  ( (
ph  /\  z : A
-1-1-onto-> C  /\  y : B -1-1-onto-> D
)  ->  C  e.  _V )
2416brrelex2i 4746 . . . . . . . . . . 11  |-  ( B 
~~  D  ->  D  e.  _V )
2519, 24syl 15 . . . . . . . . . 10  |-  ( (
ph  /\  z : A
-1-1-onto-> C  /\  y : B -1-1-onto-> D
)  ->  D  e.  _V )
2613ad2ant1 976 . . . . . . . . . 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 7415 . . . . . . . . 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 5899 . . . . . . . . . . . 12  |-  ( B  ^m  A )  e. 
_V
2928rabex 4181 . . . . . . . . . . 11  |-  { x  e.  ( B  ^m  A
)  |  ( `' x " ( _V 
\  {  .0.  }
) )  e.  Fin }  e.  _V
309, 29eqeltri 2366 . . . . . . . . . 10  |-  S  e. 
_V
3130f1oen 6898 . . . . . . . . 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 15 . . . . . . . 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 1179 . . . . . . 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 3664 . . . . . . . . . . . . . 14  |-  ( ( y `  .0.  )  =  W  ->  { ( y `  .0.  ) }  =  { W } )
3534difeq2d 3307 . . . . . . . . . . . . 13  |-  ( ( y `  .0.  )  =  W  ->  ( _V 
\  { ( y `
 .0.  ) } )  =  ( _V 
\  { W }
) )
3635imaeq2d 5028 . . . . . . . . . . . 12  |-  ( ( y `  .0.  )  =  W  ->  ( `' x " ( _V 
\  { ( y `
 .0.  ) } ) )  =  ( `' x " ( _V 
\  { W }
) ) )
3736eleq1d 2362 . . . . . . . . . . 11  |-  ( ( y `  .0.  )  =  W  ->  ( ( `' x " ( _V 
\  { ( y `
 .0.  ) } ) )  e.  Fin  <->  ( `' x " ( _V 
\  { W }
) )  e.  Fin ) )
3837rabbidv 2793 . . . . . . . . . 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 2346 . . . . . . . . 9  |-  ( ( y `  .0.  )  =  W  ->  { x  e.  ( D  ^m  C
)  |  ( `' x " ( _V 
\  { ( y `
 .0.  ) } ) )  e.  Fin }  =  T )
4140adantl 452 . . . . . . . 8  |-  ( ( y : B -1-1-onto-> D  /\  ( y `  .0.  )  =  W )  ->  { x  e.  ( D  ^m  C )  |  ( `' x " ( _V  \  {
( y `  .0.  ) } ) )  e. 
Fin }  =  T
)
42413ad2ant3 978 . . . . . . 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 4063 . . . . . 6  |-  ( (
ph  /\  z : A
-1-1-onto-> C  /\  ( y : B -1-1-onto-> D  /\  ( y `
 .0.  )  =  W ) )  ->  S  ~~  T )
44433exp 1150 . . . . 5  |-  ( ph  ->  ( z : A -1-1-onto-> C  ->  ( ( y : B -1-1-onto-> D  /\  ( y `
 .0.  )  =  W )  ->  S  ~~  T ) ) )
4544exlimdv 1626 . . . 4  |-  ( ph  ->  ( E. z  z : A -1-1-onto-> C  ->  ( (
y : B -1-1-onto-> D  /\  ( y `  .0.  )  =  W )  ->  S  ~~  T ) ) )
468, 45mpd 14 . . 3  |-  ( ph  ->  ( ( y : B -1-1-onto-> D  /\  ( y `
 .0.  )  =  W )  ->  S  ~~  T ) )
4746exlimdv 1626 . 2  |-  ( ph  ->  ( E. y ( y : B -1-1-onto-> D  /\  ( y `  .0.  )  =  W )  ->  S  ~~  T ) )
485, 47mpd 14 1  |-  ( ph  ->  S  ~~  T )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934   E.wex 1531    = wceq 1632    e. wcel 1696   {crab 2560   _Vcvv 2801    \ cdif 3162   {csn 3653   class class class wbr 4039    e. cmpt 4093   `'ccnv 4704   "cima 4708    o. ccom 4709   -1-1-onto->wf1o 5270   ` cfv 5271  (class class class)co 5874    ^m cmap 6788    ~~ cen 6876   Fincfn 6879
This theorem is referenced by:  frlmpwfi  27365
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-1o 6495  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-fin 6883
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