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Theorem dprdwd 16620
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 2830 . . 3  |-  ( ph  ->  A. x  e.  I  A  e.  ( S `  x ) )
3 eqid 2454 . . . 4  |-  ( x  e.  I  |->  A )  =  ( x  e.  I  |->  A )
43fnmpt 5648 . . 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 5893 . . . . . 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 2542 . . . 4  |-  ( (
ph  /\  x  e.  I )  ->  (
( x  e.  I  |->  A ) `  x
)  e.  ( S `
 x ) )
109ralrimiva 2830 . . 3  |-  ( ph  ->  A. x  e.  I 
( ( x  e.  I  |->  A ) `  x )  e.  ( S `  x ) )
11 nfv 1674 . . . 4  |-  F/ y ( ( x  e.  I  |->  A ) `  x )  e.  ( S `  x )
12 nffvmpt1 5810 . . . . 5  |-  F/_ x
( ( x  e.  I  |->  A ) `  y )
1312nfel1 2632 . . . 4  |-  F/ x
( ( x  e.  I  |->  A ) `  y )  e.  ( S `  y )
14 fveq2 5802 . . . . 5  |-  ( x  =  y  ->  (
( x  e.  I  |->  A ) `  x
)  =  ( ( x  e.  I  |->  A ) `  y ) )
15 fveq2 5802 . . . . 5  |-  ( x  =  y  ->  ( S `  x )  =  ( S `  y ) )
1614, 15eleq12d 2536 . . . 4  |-  ( x  =  y  ->  (
( ( x  e.  I  |->  A ) `  x )  e.  ( S `  x )  <-> 
( ( x  e.  I  |->  A ) `  y )  e.  ( S `  y ) ) )
1711, 13, 16cbvral 3049 . . 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 16619 . 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 1171 1  |-  ( ph  ->  ( x  e.  I  |->  A )  e.  W
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2799   {crab 2803   class class class wbr 4403    |-> cmpt 4461   dom cdm 4951    Fn wfn 5524   ` cfv 5529   X_cixp 7376   finSupp cfsupp 7734   DProd cdprd 16600
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 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
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 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-oprab 6207  df-mpt2 6208  df-ixp 7377  df-dprd 16602
This theorem is referenced by:  dprdfid  16632  dprdfinv  16634  dprdfadd  16635  dmdprdsplitlem  16659  dpjidcl  16682  dchrptlem3  22741
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