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Theorem mapfien2 7768
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 7529 . . 3  |-  ( (  .0.  e.  B  /\  W  e.  D  /\  B  ~~  D )  ->  E. y ( y : B -1-1-onto-> D  /\  ( y `
 .0.  )  =  W ) )
51, 2, 3, 4syl3anc 1219 . 2  |-  ( ph  ->  E. y ( y : B -1-1-onto-> D  /\  ( y `
 .0.  )  =  W ) )
6 mapfien2.ac . . . . 5  |-  ( ph  ->  A  ~~  C )
7 bren 7428 . . . . 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 2454 . . . . . . . . . 10  |-  { x  e.  ( D  ^m  C
)  |  x finSupp  (
y `  .0.  ) }  =  { x  e.  ( D  ^m  C
)  |  x finSupp  (
y `  .0.  ) }
11 eqid 2454 . . . . . . . . . 10  |-  ( y `
 .0.  )  =  ( y `  .0.  )
12 f1ocnv 5760 . . . . . . . . . . 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 7424 . . . . . . . . . . . 12  |-  Rel  ~~
1716brrelexi 4986 . . . . . . . . . . 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 4986 . . . . . . . . . . 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 4987 . . . . . . . . . . 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 4987 . . . . . . . . . . 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 7767 . . . . . . . . 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 6224 . . . . . . . . . . . 12  |-  ( B  ^m  A )  e. 
_V
2928rabex 4550 . . . . . . . . . . 11  |-  { x  e.  ( B  ^m  A
)  |  x finSupp  .0.  }  e.  _V
309, 29eqeltri 2538 . . . . . . . . . 10  |-  S  e. 
_V
3130f1oen 7439 . . . . . . . . 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 1216 . . . . . . 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 4403 . . . . . . . . . . 11  |-  ( ( y `  .0.  )  =  W  ->  ( x finSupp 
( y `  .0.  ) 
<->  x finSupp  W ) )
3534rabbidv 3068 . . . . . . . . . 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 2513 . . . . . . . . 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 4423 . . . . . 6  |-  ( (
ph  /\  z : A
-1-1-onto-> C  /\  ( y : B -1-1-onto-> D  /\  ( y `
 .0.  )  =  W ) )  ->  S  ~~  T )
41403exp 1187 . . . . 5  |-  ( ph  ->  ( z : A -1-1-onto-> C  ->  ( ( y : B -1-1-onto-> D  /\  ( y `
 .0.  )  =  W )  ->  S  ~~  T ) ) )
4241exlimdv 1691 . . . 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 1691 . 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 1370   E.wex 1587    e. wcel 1758   {crab 2802   _Vcvv 3076   class class class wbr 4399    |-> cmpt 4457   `'ccnv 4946    o. ccom 4951   -1-1-onto->wf1o 5524   ` cfv 5525  (class class class)co 6199    ^m cmap 7323    ~~ cen 7416   finSupp cfsupp 7730
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-ral 2803  df-rex 2804  df-reu 2805  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-pss 3451  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-tp 3989  df-op 3991  df-uni 4199  df-iun 4280  df-br 4400  df-opab 4458  df-mpt 4459  df-tr 4493  df-eprel 4739  df-id 4743  df-po 4748  df-so 4749  df-fr 4786  df-we 4788  df-ord 4829  df-on 4830  df-lim 4831  df-suc 4832  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-om 6586  df-1st 6686  df-2nd 6687  df-supp 6800  df-1o 7029  df-er 7210  df-map 7325  df-en 7420  df-dom 7421  df-fin 7423  df-fsupp 7731
This theorem is referenced by: (None)
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