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Theorem dprdw 17253
Description: The property of being a finitely supported function in the family  S. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 11-Jul-2019.)
Hypotheses
Ref Expression
dprdff.w  |-  W  =  { h  e.  X_ i  e.  I  ( S `  i )  |  h finSupp  .0.  }
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 finSupp  .0.  ) ) )
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 3065 . . . . 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.1 . . . . . . 7  |-  ( ph  ->  G dom DProd  S )
4 dprdff.2 . . . . . . 7  |-  ( ph  ->  dom  S  =  I )
53, 4dprddomcld 17242 . . . . . 6  |-  ( ph  ->  I  e.  _V )
6 fnex 6074 . . . . . . 7  |-  ( ( F  Fn  I  /\  I  e.  _V )  ->  F  e.  _V )
76expcom 433 . . . . . 6  |-  ( I  e.  _V  ->  ( F  Fn  I  ->  F  e.  _V ) )
85, 7syl 17 . . . . 5  |-  ( ph  ->  ( F  Fn  I  ->  F  e.  _V )
)
98adantrd 466 . . . 4  |-  ( ph  ->  ( ( F  Fn  I  /\  A. x  e.  I  ( F `  x )  e.  ( S `  x ) )  ->  F  e.  _V ) )
10 fveq2 5803 . . . . . . . . 9  |-  ( i  =  x  ->  ( S `  i )  =  ( S `  x ) )
1110cbvixpv 7443 . . . . . . . 8  |-  X_ i  e.  I  ( S `  i )  =  X_ x  e.  I  ( S `  x )
1211eleq2i 2478 . . . . . . 7  |-  ( F  e.  X_ i  e.  I 
( S `  i
)  <->  F  e.  X_ x  e.  I  ( S `  x ) )
13 elixp2 7429 . . . . . . 7  |-  ( F  e.  X_ x  e.  I 
( S `  x
)  <->  ( F  e. 
_V  /\  F  Fn  I  /\  A. x  e.  I  ( F `  x )  e.  ( S `  x ) ) )
14 3anass 976 . . . . . . 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
) ) ) )
1512, 13, 143bitri 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 ) ) ) )
1615baib 902 . . . . 5  |-  ( F  e.  _V  ->  ( F  e.  X_ i  e.  I  ( S `  i )  <->  ( F  Fn  I  /\  A. x  e.  I  ( F `  x )  e.  ( S `  x ) ) ) )
1716a1i 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 ) ) ) ) )
182, 9, 17pm5.21ndd 352 . . 3  |-  ( ph  ->  ( F  e.  X_ i  e.  I  ( S `  i )  <->  ( F  Fn  I  /\  A. x  e.  I  ( F `  x )  e.  ( S `  x ) ) ) )
1918anbi1d 703 . 2  |-  ( ph  ->  ( ( F  e.  X_ i  e.  I 
( S `  i
)  /\  F finSupp  .0.  )  <->  ( ( F  Fn  I  /\  A. x  e.  I 
( F `  x
)  e.  ( S `
 x ) )  /\  F finSupp  .0.  )
) )
20 breq1 4395 . . 3  |-  ( h  =  F  ->  (
h finSupp  .0.  <->  F finSupp  .0.  ) )
21 dprdff.w . . 3  |-  W  =  { h  e.  X_ i  e.  I  ( S `  i )  |  h finSupp  .0.  }
2220, 21elrab2 3206 . 2  |-  ( F  e.  W  <->  ( F  e.  X_ i  e.  I 
( S `  i
)  /\  F finSupp  .0.  )
)
23 df-3an 974 . 2  |-  ( ( F  Fn  I  /\  A. x  e.  I  ( F `  x )  e.  ( S `  x )  /\  F finSupp  .0.  )  <->  ( ( F  Fn  I  /\  A. x  e.  I  ( F `  x )  e.  ( S `  x
) )  /\  F finSupp  .0.  ) )
2419, 22, 233bitr4g 288 1  |-  ( ph  ->  ( F  e.  W  <->  ( F  Fn  I  /\  A. x  e.  I  ( F `  x )  e.  ( S `  x )  /\  F finSupp  .0.  ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 972    = wceq 1403    e. wcel 1840   A.wral 2751   {crab 2755   _Vcvv 3056   class class class wbr 4392   dom cdm 4940    Fn wfn 5518   ` cfv 5523   X_cixp 7425   finSupp cfsupp 7781   DProd cdprd 17234
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pr 4627  ax-un 6528
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-nel 2599  df-ral 2756  df-rex 2757  df-reu 2758  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4735  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-oprab 6236  df-mpt2 6237  df-ixp 7426  df-dprd 17236
This theorem is referenced by:  dprdwdOLD2  17255  dprdff  17256  dprdfcl  17257  dprdffsupp  17258  dprdsubg  17281
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