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Theorem dprdfcl 16614
Description: A finitely supported function in  S has its  X-th element in  S ( X ). (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 )
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 finSupp  .0.  }
3 dprdff.1 . . . . 5  |-  ( ph  ->  G dom DProd  S )
4 dprdff.2 . . . . 5  |-  ( ph  ->  dom  S  =  I )
52, 3, 4dprdw 16611 . . . 4  |-  ( ph  ->  ( F  e.  W  <->  ( F  Fn  I  /\  A. x  e.  I  ( F `  x )  e.  ( S `  x )  /\  F finSupp  .0.  ) ) )
61, 5mpbid 210 . . 3  |-  ( ph  ->  ( F  Fn  I  /\  A. x  e.  I 
( F `  x
)  e.  ( S `
 x )  /\  F finSupp  .0.  ) )
76simp2d 1001 . 2  |-  ( ph  ->  A. x  e.  I 
( F `  x
)  e.  ( S `
 x ) )
8 fveq2 5794 . . . 4  |-  ( x  =  X  ->  ( F `  x )  =  ( F `  X ) )
9 fveq2 5794 . . . 4  |-  ( x  =  X  ->  ( S `  x )  =  ( S `  X ) )
108, 9eleq12d 2534 . . 3  |-  ( x  =  X  ->  (
( F `  x
)  e.  ( S `
 x )  <->  ( F `  X )  e.  ( S `  X ) ) )
1110rspccva 3172 . 2  |-  ( ( A. x  e.  I 
( F `  x
)  e.  ( S `
 x )  /\  X  e.  I )  ->  ( F `  X
)  e.  ( S `
 X ) )
127, 11sylan 471 1  |-  ( (
ph  /\  X  e.  I )  ->  ( F `  X )  e.  ( S `  X
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   A.wral 2796   {crab 2800   class class class wbr 4395   dom cdm 4943    Fn wfn 5516   ` cfv 5521   X_cixp 7368   finSupp cfsupp 7726   DProd cdprd 16592
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pr 4634  ax-un 6477
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-reu 2803  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-oprab 6199  df-mpt2 6200  df-ixp 7369  df-dprd 16594
This theorem is referenced by:  dprdfcntz  16616  dprdfinv  16626  dprdfadd  16627  dprdfeq0  16629  dprdlub  16640  dmdprdsplitlem  16651  dpjidcl  16674
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