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Theorem dprdfcl 16849
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 16846 . . . 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 1009 . 2  |-  ( ph  ->  A. x  e.  I 
( F `  x
)  e.  ( S `
 x ) )
8 fveq2 5866 . . . 4  |-  ( x  =  X  ->  ( F `  x )  =  ( F `  X ) )
9 fveq2 5866 . . . 4  |-  ( x  =  X  ->  ( S `  x )  =  ( S `  X ) )
108, 9eleq12d 2549 . . 3  |-  ( x  =  X  ->  (
( F `  x
)  e.  ( S `
 x )  <->  ( F `  X )  e.  ( S `  X ) ) )
1110rspccva 3213 . 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 973    = wceq 1379    e. wcel 1767   A.wral 2814   {crab 2818   class class class wbr 4447   dom cdm 4999    Fn wfn 5583   ` cfv 5588   X_cixp 7469   finSupp cfsupp 7829   DProd cdprd 16827
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 6576
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 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-oprab 6288  df-mpt2 6289  df-ixp 7470  df-dprd 16829
This theorem is referenced by:  dprdfcntz  16851  dprdfinv  16861  dprdfadd  16862  dprdfeq0  16864  dprdlub  16875  dmdprdsplitlem  16886  dpjidcl  16909
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