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Theorem dprdfcl 15248
Description: A finitely supported function in  S has its  X-th element in  S ( X ). (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 )
dprdff.3  |-  ( ph  ->  F  e.  W )
Assertion
Ref Expression
dprdfcl  |-  ( (
ph  /\  X  e.  I )  ->  ( F `  X )  e.  ( S `  X
) )
Distinct variable groups:    h, F    h, i, I    .0. , h    S, h, i
Allowed substitution hints:    ph( h, i)    F( i)    G( h, i)    W( h, i)    X( h, i)    .0. ( i)

Proof of Theorem dprdfcl
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 dprdff.3 . . . 4  |-  ( ph  ->  F  e.  W )
2 dprdff.w . . . . 5  |-  W  =  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V  \  {  .0.  } ) )  e. 
Fin }
3 dprdff.1 . . . . 5  |-  ( ph  ->  G dom DProd  S )
4 dprdff.2 . . . . 5  |-  ( ph  ->  dom  S  =  I )
52, 3, 4dprdw 15245 . . . 4  |-  ( ph  ->  ( F  e.  W  <->  ( F  Fn  I  /\  A. x  e.  I  ( F `  x )  e.  ( S `  x )  /\  ( `' F " ( _V 
\  {  .0.  }
) )  e.  Fin ) ) )
61, 5mpbid 201 . . 3  |-  ( ph  ->  ( F  Fn  I  /\  A. x  e.  I 
( F `  x
)  e.  ( S `
 x )  /\  ( `' F " ( _V 
\  {  .0.  }
) )  e.  Fin ) )
76simp2d 968 . 2  |-  ( ph  ->  A. x  e.  I 
( F `  x
)  e.  ( S `
 x ) )
8 fveq2 5525 . . . 4  |-  ( x  =  X  ->  ( F `  x )  =  ( F `  X ) )
9 fveq2 5525 . . . 4  |-  ( x  =  X  ->  ( S `  x )  =  ( S `  X ) )
108, 9eleq12d 2351 . . 3  |-  ( x  =  X  ->  (
( F `  x
)  e.  ( S `
 x )  <->  ( F `  X )  e.  ( S `  X ) ) )
1110rspccva 2883 . 2  |-  ( ( A. x  e.  I 
( F `  x
)  e.  ( S `
 x )  /\  X  e.  I )  ->  ( F `  X
)  e.  ( S `
 X ) )
127, 11sylan 457 1  |-  ( (
ph  /\  X  e.  I )  ->  ( F `  X )  e.  ( S `  X
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   {crab 2547   _Vcvv 2788    \ cdif 3149   {csn 3640   class class class wbr 4023   `'ccnv 4688   dom cdm 4689   "cima 4692    Fn wfn 5250   ` cfv 5255   X_cixp 6817   Fincfn 6863   DProd cdprd 15231
This theorem is referenced by:  dprdfcntz  15250  dprdfinv  15254  dprdfadd  15255  dprdfeq0  15257  dprdlub  15261  dmdprdsplitlem  15272  dpjidcl  15293
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-oprab 5862  df-mpt2 5863  df-ixp 6818  df-dprd 15233
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