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Theorem dprdwd 16917
Description: A mapping 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 )
dprdwd.3  |-  ( (
ph  /\  x  e.  I )  ->  A  e.  ( S `  x
) )
dprdwd.4  |-  ( ph  ->  ( x  e.  I  |->  A ) finSupp  .0.  )
Assertion
Ref Expression
dprdwd  |-  ( ph  ->  ( x  e.  I  |->  A )  e.  W
)
Distinct variable groups:    A, h    x, h    x, G    h, i, I, x    .0. , h    ph, x    S, h, i, x
Allowed substitution hints:    ph( h, i)    A( x, i)    G( h, i)    W( x, h, i)    .0. ( x, i)

Proof of Theorem dprdwd
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 dprdwd.3 . . . 4  |-  ( (
ph  /\  x  e.  I )  ->  A  e.  ( S `  x
) )
21ralrimiva 2881 . . 3  |-  ( ph  ->  A. x  e.  I  A  e.  ( S `  x ) )
3 eqid 2467 . . . 4  |-  ( x  e.  I  |->  A )  =  ( x  e.  I  |->  A )
43fnmpt 5713 . . 3  |-  ( A. x  e.  I  A  e.  ( S `  x
)  ->  ( x  e.  I  |->  A )  Fn  I )
52, 4syl 16 . 2  |-  ( ph  ->  ( x  e.  I  |->  A )  Fn  I
)
6 simpr 461 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  x  e.  I )
73fvmpt2 5964 . . . . . 6  |-  ( ( x  e.  I  /\  A  e.  ( S `  x ) )  -> 
( ( x  e.  I  |->  A ) `  x )  =  A )
86, 1, 7syl2anc 661 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  (
( x  e.  I  |->  A ) `  x
)  =  A )
98, 1eqeltrd 2555 . . . 4  |-  ( (
ph  /\  x  e.  I )  ->  (
( x  e.  I  |->  A ) `  x
)  e.  ( S `
 x ) )
109ralrimiva 2881 . . 3  |-  ( ph  ->  A. x  e.  I 
( ( x  e.  I  |->  A ) `  x )  e.  ( S `  x ) )
11 nfv 1683 . . . 4  |-  F/ y ( ( x  e.  I  |->  A ) `  x )  e.  ( S `  x )
12 nffvmpt1 5880 . . . . 5  |-  F/_ x
( ( x  e.  I  |->  A ) `  y )
1312nfel1 2645 . . . 4  |-  F/ x
( ( x  e.  I  |->  A ) `  y )  e.  ( S `  y )
14 fveq2 5872 . . . . 5  |-  ( x  =  y  ->  (
( x  e.  I  |->  A ) `  x
)  =  ( ( x  e.  I  |->  A ) `  y ) )
15 fveq2 5872 . . . . 5  |-  ( x  =  y  ->  ( S `  x )  =  ( S `  y ) )
1614, 15eleq12d 2549 . . . 4  |-  ( x  =  y  ->  (
( ( x  e.  I  |->  A ) `  x )  e.  ( S `  x )  <-> 
( ( x  e.  I  |->  A ) `  y )  e.  ( S `  y ) ) )
1711, 13, 16cbvral 3089 . . 3  |-  ( A. x  e.  I  (
( x  e.  I  |->  A ) `  x
)  e.  ( S `
 x )  <->  A. y  e.  I  ( (
x  e.  I  |->  A ) `  y )  e.  ( S `  y ) )
1810, 17sylib 196 . 2  |-  ( ph  ->  A. y  e.  I 
( ( x  e.  I  |->  A ) `  y )  e.  ( S `  y ) )
19 dprdwd.4 . 2  |-  ( ph  ->  ( x  e.  I  |->  A ) finSupp  .0.  )
20 dprdff.w . . 3  |-  W  =  { h  e.  X_ i  e.  I  ( S `  i )  |  h finSupp  .0.  }
21 dprdff.1 . . 3  |-  ( ph  ->  G dom DProd  S )
22 dprdff.2 . . 3  |-  ( ph  ->  dom  S  =  I )
2320, 21, 22dprdw 16916 . 2  |-  ( ph  ->  ( ( x  e.  I  |->  A )  e.  W  <->  ( ( x  e.  I  |->  A )  Fn  I  /\  A. y  e.  I  (
( x  e.  I  |->  A ) `  y
)  e.  ( S `
 y )  /\  ( x  e.  I  |->  A ) finSupp  .0.  )
) )
245, 18, 19, 23mpbir3and 1179 1  |-  ( ph  ->  ( x  e.  I  |->  A )  e.  W
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2817   {crab 2821   class class class wbr 4453    |-> cmpt 4511   dom cdm 5005    Fn wfn 5589   ` cfv 5594   X_cixp 7481   finSupp cfsupp 7841   DProd cdprd 16897
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 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
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 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-oprab 6299  df-mpt2 6300  df-ixp 7482  df-dprd 16899
This theorem is referenced by:  dprdfid  16929  dprdfinv  16931  dprdfadd  16932  dmdprdsplitlem  16956  dpjidcl  16979  dchrptlem3  23407
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