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Theorem dprdwOLD 16840
Description: The property of being a finitely supported function in the family  S. (Contributed by Mario Carneiro, 25-Apr-2016.) Obsolete version of dprdw 16834 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 3122 . . . . 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 16819 . . . . . . . . 9  |-  Rel  dom DProd
65brrelex2i 5040 . . . . . . . 8  |-  ( G dom DProd  S  ->  S  e. 
_V )
7 dmexg 6712 . . . . . . . 8  |-  ( S  e.  _V  ->  dom  S  e.  _V )
84, 6, 73syl 20 . . . . . . 7  |-  ( ph  ->  dom  S  e.  _V )
93, 8eqeltrrd 2556 . . . . . 6  |-  ( ph  ->  I  e.  _V )
10 fnex 6125 . . . . . . 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 5864 . . . . . . . . 9  |-  ( i  =  x  ->  ( S `  i )  =  ( S `  x ) )
1514cbvixpv 7484 . . . . . . . 8  |-  X_ i  e.  I  ( S `  i )  =  X_ x  e.  I  ( S `  x )
1615eleq2i 2545 . . . . . . 7  |-  ( F  e.  X_ i  e.  I 
( S `  i
)  <->  F  e.  X_ x  e.  I  ( S `  x ) )
17 elixp2 7470 . . . . . . 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 901 . . . . 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 5174 . . . . 5  |-  ( h  =  F  ->  `' h  =  `' F
)
2524imaeq1d 5334 . . . 4  |-  ( h  =  F  ->  ( `' h " ( _V 
\  {  .0.  }
) )  =  ( `' F " ( _V 
\  {  .0.  }
) ) )
2625eleq1d 2536 . . 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 3263 . 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 1379    e. wcel 1767   A.wral 2814   {crab 2818   _Vcvv 3113    \ cdif 3473   {csn 4027   class class class wbr 4447   `'ccnv 4998   dom cdm 4999   "cima 5002    Fn wfn 5581   ` cfv 5586   X_cixp 7466   Fincfn 7513   DProd cdprd 16815
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-oprab 6286  df-mpt2 6287  df-ixp 7467  df-dprd 16817
This theorem is referenced by:  dprdwdOLD  16841  dprdffOLD  16842  dprdfclOLD  16843  dprdffiOLD  16844
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