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Theorem dprdw 15497
Description: The property of being a finitely supported function in the family  S. (Contributed by Mario Carneiro, 25-Apr-2016.)
Hypotheses
Ref Expression
dprdff.w  |-  W  =  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V  \  {  .0.  } ) )  e. 
Fin }
dprdff.1  |-  ( ph  ->  G dom DProd  S )
dprdff.2  |-  ( ph  ->  dom  S  =  I )
Assertion
Ref Expression
dprdw  |-  ( 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 dprdw
StepHypRef Expression
1 elex 2909 . . . . 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 dprdff.2 . . . . . . 7  |-  ( ph  ->  dom  S  =  I )
4 dprdff.1 . . . . . . . 8  |-  ( ph  ->  G dom DProd  S )
5 reldmdprd 15487 . . . . . . . . 9  |-  Rel  dom DProd
65brrelex2i 4861 . . . . . . . 8  |-  ( G dom DProd  S  ->  S  e. 
_V )
7 dmexg 5072 . . . . . . . 8  |-  ( S  e.  _V  ->  dom  S  e.  _V )
84, 6, 73syl 19 . . . . . . 7  |-  ( ph  ->  dom  S  e.  _V )
93, 8eqeltrrd 2464 . . . . . 6  |-  ( ph  ->  I  e.  _V )
10 fnex 5902 . . . . . . 7  |-  ( ( F  Fn  I  /\  I  e.  _V )  ->  F  e.  _V )
1110expcom 425 . . . . . 6  |-  ( I  e.  _V  ->  ( F  Fn  I  ->  F  e.  _V ) )
129, 11syl 16 . . . . 5  |-  ( ph  ->  ( F  Fn  I  ->  F  e.  _V )
)
1312adantrd 455 . . . 4  |-  ( ph  ->  ( ( F  Fn  I  /\  A. x  e.  I  ( F `  x )  e.  ( S `  x ) )  ->  F  e.  _V ) )
14 fveq2 5670 . . . . . . . . 9  |-  ( i  =  x  ->  ( S `  i )  =  ( S `  x ) )
1514cbvixpv 7018 . . . . . . . 8  |-  X_ i  e.  I  ( S `  i )  =  X_ x  e.  I  ( S `  x )
1615eleq2i 2453 . . . . . . 7  |-  ( F  e.  X_ i  e.  I 
( S `  i
)  <->  F  e.  X_ x  e.  I  ( S `  x ) )
17 elixp2 7004 . . . . . . 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 940 . . . . . . 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 263 . . . . . 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 872 . . . . 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 344 . . 3  |-  ( ph  ->  ( F  e.  X_ i  e.  I  ( S `  i )  <->  ( F  Fn  I  /\  A. x  e.  I  ( F `  x )  e.  ( S `  x ) ) ) )
2322anbi1d 686 . 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 4988 . . . . 5  |-  ( h  =  F  ->  `' h  =  `' F
)
2524imaeq1d 5144 . . . 4  |-  ( h  =  F  ->  ( `' h " ( _V 
\  {  .0.  }
) )  =  ( `' F " ( _V 
\  {  .0.  }
) ) )
2625eleq1d 2455 . . 3  |-  ( h  =  F  ->  (
( `' h "
( _V  \  {  .0.  } ) )  e. 
Fin 
<->  ( `' F "
( _V  \  {  .0.  } ) )  e. 
Fin ) )
27 dprdff.w . . 3  |-  W  =  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V  \  {  .0.  } ) )  e. 
Fin }
2826, 27elrab2 3039 . 2  |-  ( F  e.  W  <->  ( F  e.  X_ i  e.  I 
( S `  i
)  /\  ( `' F " ( _V  \  {  .0.  } ) )  e.  Fin ) )
29 df-3an 938 . 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 280 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 set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   A.wral 2651   {crab 2655   _Vcvv 2901    \ cdif 3262   {csn 3759   class class class wbr 4155   `'ccnv 4819   dom cdm 4820   "cima 4823    Fn wfn 5391   ` cfv 5396   X_cixp 7001   Fincfn 7047   DProd cdprd 15483
This theorem is referenced by:  dprdwd  15498  dprdff  15499  dprdfcl  15500  dprdffi  15501  dprdsubg  15511
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-oprab 6026  df-mpt2 6027  df-ixp 7002  df-dprd 15485
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