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Theorem dprdwOLD 17176
Description: The property of being a finitely supported function in the family  S. (Contributed by Mario Carneiro, 25-Apr-2016.) Obsolete version of dprdw 17169 as of 11-Jul-2019. (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
dprdffOLD.w  |-  W  =  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V  \  {  .0.  } ) )  e. 
Fin }
dprdffOLD.1  |-  ( ph  ->  G dom DProd  S )
dprdffOLD.2  |-  ( ph  ->  dom  S  =  I )
Assertion
Ref Expression
dprdwOLD  |-  ( ph  ->  ( F  e.  W  <->  ( F  Fn  I  /\  A. x  e.  I  ( F `  x )  e.  ( S `  x )  /\  ( `' F " ( _V 
\  {  .0.  }
) )  e.  Fin ) ) )
Distinct variable groups:    x, h, F    x, G    h, i, I, x    .0. , h    ph, x    S, h, i, x
Allowed substitution hints:    ph( h, i)    F( i)    G( h, i)    W( x, h, i)    .0. ( x, i)

Proof of Theorem dprdwOLD
StepHypRef Expression
1 elex 3118 . . . . 5  |-  ( F  e.  X_ i  e.  I 
( S `  i
)  ->  F  e.  _V )
21a1i 11 . . . 4  |-  ( ph  ->  ( F  e.  X_ i  e.  I  ( S `  i )  ->  F  e.  _V )
)
3 dprdffOLD.2 . . . . . . 7  |-  ( ph  ->  dom  S  =  I )
4 dprdffOLD.1 . . . . . . . 8  |-  ( ph  ->  G dom DProd  S )
5 reldmdprd 17154 . . . . . . . . 9  |-  Rel  dom DProd
65brrelex2i 5050 . . . . . . . 8  |-  ( G dom DProd  S  ->  S  e. 
_V )
7 dmexg 6730 . . . . . . . 8  |-  ( S  e.  _V  ->  dom  S  e.  _V )
84, 6, 73syl 20 . . . . . . 7  |-  ( ph  ->  dom  S  e.  _V )
93, 8eqeltrrd 2546 . . . . . 6  |-  ( ph  ->  I  e.  _V )
10 fnex 6140 . . . . . . 7  |-  ( ( F  Fn  I  /\  I  e.  _V )  ->  F  e.  _V )
1110expcom 435 . . . . . 6  |-  ( I  e.  _V  ->  ( F  Fn  I  ->  F  e.  _V ) )
129, 11syl 16 . . . . 5  |-  ( ph  ->  ( F  Fn  I  ->  F  e.  _V )
)
1312adantrd 468 . . . 4  |-  ( ph  ->  ( ( F  Fn  I  /\  A. x  e.  I  ( F `  x )  e.  ( S `  x ) )  ->  F  e.  _V ) )
14 fveq2 5872 . . . . . . . . 9  |-  ( i  =  x  ->  ( S `  i )  =  ( S `  x ) )
1514cbvixpv 7506 . . . . . . . 8  |-  X_ i  e.  I  ( S `  i )  =  X_ x  e.  I  ( S `  x )
1615eleq2i 2535 . . . . . . 7  |-  ( F  e.  X_ i  e.  I 
( S `  i
)  <->  F  e.  X_ x  e.  I  ( S `  x ) )
17 elixp2 7492 . . . . . . 7  |-  ( F  e.  X_ x  e.  I 
( S `  x
)  <->  ( F  e. 
_V  /\  F  Fn  I  /\  A. x  e.  I  ( F `  x )  e.  ( S `  x ) ) )
18 3anass 977 . . . . . . 7  |-  ( ( F  e.  _V  /\  F  Fn  I  /\  A. x  e.  I  ( F `  x )  e.  ( S `  x ) )  <->  ( F  e.  _V  /\  ( F  Fn  I  /\  A. x  e.  I  ( F `  x )  e.  ( S `  x
) ) ) )
1916, 17, 183bitri 271 . . . . . 6  |-  ( F  e.  X_ i  e.  I 
( S `  i
)  <->  ( F  e. 
_V  /\  ( F  Fn  I  /\  A. x  e.  I  ( F `  x )  e.  ( S `  x ) ) ) )
2019baib 903 . . . . 5  |-  ( F  e.  _V  ->  ( F  e.  X_ i  e.  I  ( S `  i )  <->  ( F  Fn  I  /\  A. x  e.  I  ( F `  x )  e.  ( S `  x ) ) ) )
2120a1i 11 . . . 4  |-  ( ph  ->  ( F  e.  _V  ->  ( F  e.  X_ i  e.  I  ( S `  i )  <->  ( F  Fn  I  /\  A. x  e.  I  ( F `  x )  e.  ( S `  x ) ) ) ) )
222, 13, 21pm5.21ndd 354 . . 3  |-  ( ph  ->  ( F  e.  X_ i  e.  I  ( S `  i )  <->  ( F  Fn  I  /\  A. x  e.  I  ( F `  x )  e.  ( S `  x ) ) ) )
2322anbi1d 704 . 2  |-  ( ph  ->  ( ( F  e.  X_ i  e.  I 
( S `  i
)  /\  ( `' F " ( _V  \  {  .0.  } ) )  e.  Fin )  <->  ( ( F  Fn  I  /\  A. x  e.  I  ( F `  x )  e.  ( S `  x ) )  /\  ( `' F " ( _V 
\  {  .0.  }
) )  e.  Fin ) ) )
24 cnveq 5186 . . . . 5  |-  ( h  =  F  ->  `' h  =  `' F
)
2524imaeq1d 5346 . . . 4  |-  ( h  =  F  ->  ( `' h " ( _V 
\  {  .0.  }
) )  =  ( `' F " ( _V 
\  {  .0.  }
) ) )
2625eleq1d 2526 . . 3  |-  ( h  =  F  ->  (
( `' h "
( _V  \  {  .0.  } ) )  e. 
Fin 
<->  ( `' F "
( _V  \  {  .0.  } ) )  e. 
Fin ) )
27 dprdffOLD.w . . 3  |-  W  =  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V  \  {  .0.  } ) )  e. 
Fin }
2826, 27elrab2 3259 . 2  |-  ( F  e.  W  <->  ( F  e.  X_ i  e.  I 
( S `  i
)  /\  ( `' F " ( _V  \  {  .0.  } ) )  e.  Fin ) )
29 df-3an 975 . 2  |-  ( ( F  Fn  I  /\  A. x  e.  I  ( F `  x )  e.  ( S `  x )  /\  ( `' F " ( _V 
\  {  .0.  }
) )  e.  Fin ) 
<->  ( ( F  Fn  I  /\  A. x  e.  I  ( F `  x )  e.  ( S `  x ) )  /\  ( `' F " ( _V 
\  {  .0.  }
) )  e.  Fin ) )
3023, 28, 293bitr4g 288 1  |-  ( ph  ->  ( F  e.  W  <->  ( F  Fn  I  /\  A. x  e.  I  ( F `  x )  e.  ( S `  x )  /\  ( `' F " ( _V 
\  {  .0.  }
) )  e.  Fin ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819   A.wral 2807   {crab 2811   _Vcvv 3109    \ cdif 3468   {csn 4032   class class class wbr 4456   `'ccnv 5007   dom cdm 5008   "cima 5011    Fn wfn 5589   ` cfv 5594   X_cixp 7488   Fincfn 7535   DProd cdprd 17150
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-oprab 6300  df-mpt2 6301  df-ixp 7489  df-dprd 17152
This theorem is referenced by:  dprdwdOLD  17177  dprdffOLD  17178  dprdfclOLD  17179  dprdffiOLD  17180
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