Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  mapfien2OLD Structured version   Unicode version

Theorem mapfien2OLD 29472
Description: Equinumerousity relation for sets of finitely supported functions. MOVABLE (Contributed by Stefan O'Rear, 9-Jul-2015.) Obsolete version of mapfien2 7677 as of 7-Jul-2019. (New usage is discouraged.)
Hypotheses
Ref Expression
mapfien2OLD.s  |-  S  =  { x  e.  ( B  ^m  A )  |  ( `' x " ( _V  \  {  .0.  } ) )  e. 
Fin }
mapfien2OLD.t  |-  T  =  { x  e.  ( D  ^m  C )  |  ( `' x " ( _V  \  { W } ) )  e. 
Fin }
mapfien2OLD.ac  |-  ( ph  ->  A  ~~  C )
mapfien2OLD.bd  |-  ( ph  ->  B  ~~  D )
mapfien2OLD.z  |-  ( ph  ->  .0.  e.  B )
mapfien2OLD.w  |-  ( ph  ->  W  e.  D )
Assertion
Ref Expression
mapfien2OLD  |-  ( 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 mapfien2OLD
Dummy variables  w  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mapfien2OLD.z . . 3  |-  ( ph  ->  .0.  e.  B )
2 mapfien2OLD.w . . 3  |-  ( ph  ->  W  e.  D )
3 mapfien2OLD.bd . . 3  |-  ( ph  ->  B  ~~  D )
4 enfixsn 7439 . . 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 mapfien2OLD.ac . . . . 5  |-  ( ph  ->  A  ~~  C )
7 bren 7338 . . . . 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 mapfien2OLD.s . . . . . . . . . 10  |-  S  =  { x  e.  ( B  ^m  A )  |  ( `' x " ( _V  \  {  .0.  } ) )  e. 
Fin }
10 eqid 2443 . . . . . . . . . 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 2443 . . . . . . . . . 10  |-  ( y `
 .0.  )  =  ( y `  .0.  )
12 f1ocnv 5672 . . . . . . . . . . 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 7334 . . . . . . . . . . . 12  |-  Rel  ~~
1716brrelexi 4898 . . . . . . . . . . 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 4898 . . . . . . . . . . 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 4899 . . . . . . . . . . 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 4899 . . . . . . . . . . 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, 26mapfienOLD 7946 . . . . . . . . 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 6135 . . . . . . . . . . . 12  |-  ( B  ^m  A )  e. 
_V
2928rabex 4462 . . . . . . . . . . 11  |-  { x  e.  ( B  ^m  A
)  |  ( `' x " ( _V 
\  {  .0.  }
) )  e.  Fin }  e.  _V
309, 29eqeltri 2513 . . . . . . . . . 10  |-  S  e. 
_V
3130f1oen 7349 . . . . . . . . 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 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 " ( _V 
\  { ( y `
 .0.  ) } ) )  e.  Fin } )
34 sneq 3906 . . . . . . . . . . . . . 14  |-  ( ( y `  .0.  )  =  W  ->  { ( y `  .0.  ) }  =  { W } )
3534difeq2d 3493 . . . . . . . . . . . . 13  |-  ( ( y `  .0.  )  =  W  ->  ( _V 
\  { ( y `
 .0.  ) } )  =  ( _V 
\  { W }
) )
3635imaeq2d 5188 . . . . . . . . . . . 12  |-  ( ( y `  .0.  )  =  W  ->  ( `' x " ( _V 
\  { ( y `
 .0.  ) } ) )  =  ( `' x " ( _V 
\  { W }
) ) )
3736eleq1d 2509 . . . . . . . . . . 11  |-  ( ( y `  .0.  )  =  W  ->  ( ( `' x " ( _V 
\  { ( y `
 .0.  ) } ) )  e.  Fin  <->  ( `' x " ( _V 
\  { W }
) )  e.  Fin ) )
3837rabbidv 2983 . . . . . . . . . 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 mapfien2OLD.t . . . . . . . . . 10  |-  T  =  { x  e.  ( D  ^m  C )  |  ( `' x " ( _V  \  { W } ) )  e. 
Fin }
4038, 39syl6eqr 2493 . . . . . . . . 9  |-  ( ( y `  .0.  )  =  W  ->  { x  e.  ( D  ^m  C
)  |  ( `' x " ( _V 
\  { ( y `
 .0.  ) } ) )  e.  Fin }  =  T )
4140adantl 466 . . . . . . . 8  |-  ( ( y : B -1-1-onto-> D  /\  ( y `  .0.  )  =  W )  ->  { x  e.  ( D  ^m  C )  |  ( `' x " ( _V  \  {
( y `  .0.  ) } ) )  e. 
Fin }  =  T
)
42413ad2ant3 1011 . . . . . . 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 4335 . . . . . 6  |-  ( (
ph  /\  z : A
-1-1-onto-> C  /\  ( y : B -1-1-onto-> D  /\  ( y `
 .0.  )  =  W ) )  ->  S  ~~  T )
44433exp 1186 . . . . 5  |-  ( ph  ->  ( z : A -1-1-onto-> C  ->  ( ( y : B -1-1-onto-> D  /\  ( y `
 .0.  )  =  W )  ->  S  ~~  T ) ) )
4544exlimdv 1690 . . . 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 1690 . 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369   E.wex 1586    e. wcel 1756   {crab 2738   _Vcvv 2991    \ cdif 3344   {csn 3896   class class class wbr 4311    e. cmpt 4369   `'ccnv 4858   "cima 4862    o. ccom 4863   -1-1-onto->wf1o 5436   ` cfv 5437  (class class class)co 6110    ^m cmap 7233    ~~ cen 7326   Fincfn 7329
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 2423  ax-rep 4422  ax-sep 4432  ax-nul 4440  ax-pow 4489  ax-pr 4550  ax-un 6391
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 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2739  df-rex 2740  df-reu 2741  df-rab 2743  df-v 2993  df-sbc 3206  df-csb 3308  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-pss 3363  df-nul 3657  df-if 3811  df-pw 3881  df-sn 3897  df-pr 3899  df-tp 3901  df-op 3903  df-uni 4111  df-iun 4192  df-br 4312  df-opab 4370  df-mpt 4371  df-tr 4405  df-eprel 4651  df-id 4655  df-po 4660  df-so 4661  df-fr 4698  df-we 4700  df-ord 4741  df-on 4742  df-lim 4743  df-suc 4744  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-rn 4870  df-res 4871  df-ima 4872  df-iota 5400  df-fun 5439  df-fn 5440  df-f 5441  df-f1 5442  df-fo 5443  df-f1o 5444  df-fv 5445  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-om 6496  df-1st 6596  df-2nd 6597  df-1o 6939  df-er 7120  df-map 7235  df-en 7330  df-dom 7331  df-fin 7333
This theorem is referenced by:  frlmpwfi  29476
  Copyright terms: Public domain W3C validator