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Theorem dprdw 15560
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 2956 . . . . 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 15550 . . . . . . . . 9  |-  Rel  dom DProd
65brrelex2i 4911 . . . . . . . 8  |-  ( G dom DProd  S  ->  S  e. 
_V )
7 dmexg 5122 . . . . . . . 8  |-  ( S  e.  _V  ->  dom  S  e.  _V )
84, 6, 73syl 19 . . . . . . 7  |-  ( ph  ->  dom  S  e.  _V )
93, 8eqeltrrd 2510 . . . . . 6  |-  ( ph  ->  I  e.  _V )
10 fnex 5953 . . . . . . 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 5720 . . . . . . . . 9  |-  ( i  =  x  ->  ( S `  i )  =  ( S `  x ) )
1514cbvixpv 7072 . . . . . . . 8  |-  X_ i  e.  I  ( S `  i )  =  X_ x  e.  I  ( S `  x )
1615eleq2i 2499 . . . . . . 7  |-  ( F  e.  X_ i  e.  I 
( S `  i
)  <->  F  e.  X_ x  e.  I  ( S `  x ) )
17 elixp2 7058 . . . . . . 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 5038 . . . . 5  |-  ( h  =  F  ->  `' h  =  `' F
)
2524imaeq1d 5194 . . . 4  |-  ( h  =  F  ->  ( `' h " ( _V 
\  {  .0.  }
) )  =  ( `' F " ( _V 
\  {  .0.  }
) ) )
2625eleq1d 2501 . . 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 3086 . 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 1652    e. wcel 1725   A.wral 2697   {crab 2701   _Vcvv 2948    \ cdif 3309   {csn 3806   class class class wbr 4204   `'ccnv 4869   dom cdm 4870   "cima 4873    Fn wfn 5441   ` cfv 5446   X_cixp 7055   Fincfn 7101   DProd cdprd 15546
This theorem is referenced by:  dprdwd  15561  dprdff  15562  dprdfcl  15563  dprdffi  15564  dprdsubg  15574
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-oprab 6077  df-mpt2 6078  df-ixp 7056  df-dprd 15548
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