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Theorem dprdwd 15498
Description: The property of being a finitely supported function in the family  S. (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 )
dprdwd.3  |-  ( (
ph  /\  x  e.  I )  ->  A  e.  ( S `  x
) )
dprdwd.4  |-  ( ph  ->  ( `' ( x  e.  I  |->  A )
" ( _V  \  {  .0.  } ) )  e.  Fin )
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 2734 . . 3  |-  ( ph  ->  A. x  e.  I  A  e.  ( S `  x ) )
3 eqid 2389 . . . 4  |-  ( x  e.  I  |->  A )  =  ( x  e.  I  |->  A )
43fnmpt 5513 . . 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 448 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  x  e.  I )
73fvmpt2 5753 . . . . . 6  |-  ( ( x  e.  I  /\  A  e.  ( S `  x ) )  -> 
( ( x  e.  I  |->  A ) `  x )  =  A )
86, 1, 7syl2anc 643 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  (
( x  e.  I  |->  A ) `  x
)  =  A )
98, 1eqeltrd 2463 . . . 4  |-  ( (
ph  /\  x  e.  I )  ->  (
( x  e.  I  |->  A ) `  x
)  e.  ( S `
 x ) )
109ralrimiva 2734 . . 3  |-  ( ph  ->  A. x  e.  I 
( ( x  e.  I  |->  A ) `  x )  e.  ( S `  x ) )
11 nfv 1626 . . . 4  |-  F/ y ( ( x  e.  I  |->  A ) `  x )  e.  ( S `  x )
12 nffvmpt1 5678 . . . . 5  |-  F/_ x
( ( x  e.  I  |->  A ) `  y )
1312nfel1 2535 . . . 4  |-  F/ x
( ( x  e.  I  |->  A ) `  y )  e.  ( S `  y )
14 fveq2 5670 . . . . 5  |-  ( x  =  y  ->  (
( x  e.  I  |->  A ) `  x
)  =  ( ( x  e.  I  |->  A ) `  y ) )
15 fveq2 5670 . . . . 5  |-  ( x  =  y  ->  ( S `  x )  =  ( S `  y ) )
1614, 15eleq12d 2457 . . . 4  |-  ( x  =  y  ->  (
( ( x  e.  I  |->  A ) `  x )  e.  ( S `  x )  <-> 
( ( x  e.  I  |->  A ) `  y )  e.  ( S `  y ) ) )
1711, 13, 16cbvral 2873 . . 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 189 . 2  |-  ( ph  ->  A. y  e.  I 
( ( x  e.  I  |->  A ) `  y )  e.  ( S `  y ) )
19 dprdwd.4 . 2  |-  ( ph  ->  ( `' ( x  e.  I  |->  A )
" ( _V  \  {  .0.  } ) )  e.  Fin )
20 dprdff.w . . 3  |-  W  =  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V  \  {  .0.  } ) )  e. 
Fin }
21 dprdff.1 . . 3  |-  ( ph  ->  G dom DProd  S )
22 dprdff.2 . . 3  |-  ( ph  ->  dom  S  =  I )
2320, 21, 22dprdw 15497 . 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 ) "
( _V  \  {  .0.  } ) )  e. 
Fin ) ) )
245, 18, 19, 23mpbir3and 1137 1  |-  ( ph  ->  ( x  e.  I  |->  A )  e.  W
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2651   {crab 2655   _Vcvv 2901    \ cdif 3262   {csn 3759   class class class wbr 4155    e. cmpt 4209   `'ccnv 4819   dom cdm 4820   "cima 4823    Fn wfn 5391   ` cfv 5396   X_cixp 7001   Fincfn 7047   DProd cdprd 15483
This theorem is referenced by:  dprdfid  15504  dprdfinv  15506  dprdfadd  15507  dmdprdsplitlem  15524  dpjidcl  15545  dchrptlem3  20919
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-oprab 6026  df-mpt2 6027  df-ixp 7002  df-dprd 15485
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