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Theorem dprdwd 15262
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 2639 . . 3  |-  ( ph  ->  A. x  e.  I  A  e.  ( S `  x ) )
3 eqid 2296 . . . 4  |-  ( x  e.  I  |->  A )  =  ( x  e.  I  |->  A )
43fnmpt 5386 . . 3  |-  ( A. x  e.  I  A  e.  ( S `  x
)  ->  ( x  e.  I  |->  A )  Fn  I )
52, 4syl 15 . 2  |-  ( ph  ->  ( x  e.  I  |->  A )  Fn  I
)
6 simpr 447 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  x  e.  I )
73fvmpt2 5624 . . . . . 6  |-  ( ( x  e.  I  /\  A  e.  ( S `  x ) )  -> 
( ( x  e.  I  |->  A ) `  x )  =  A )
86, 1, 7syl2anc 642 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  (
( x  e.  I  |->  A ) `  x
)  =  A )
98, 1eqeltrd 2370 . . . 4  |-  ( (
ph  /\  x  e.  I )  ->  (
( x  e.  I  |->  A ) `  x
)  e.  ( S `
 x ) )
109ralrimiva 2639 . . 3  |-  ( ph  ->  A. x  e.  I 
( ( x  e.  I  |->  A ) `  x )  e.  ( S `  x ) )
11 nfv 1609 . . . 4  |-  F/ y ( ( x  e.  I  |->  A ) `  x )  e.  ( S `  x )
12 nfmpt1 4125 . . . . . 6  |-  F/_ x
( x  e.  I  |->  A )
13 nfcv 2432 . . . . . 6  |-  F/_ x
y
1412, 13nffv 5548 . . . . 5  |-  F/_ x
( ( x  e.  I  |->  A ) `  y )
1514nfel1 2442 . . . 4  |-  F/ x
( ( x  e.  I  |->  A ) `  y )  e.  ( S `  y )
16 fveq2 5541 . . . . 5  |-  ( x  =  y  ->  (
( x  e.  I  |->  A ) `  x
)  =  ( ( x  e.  I  |->  A ) `  y ) )
17 fveq2 5541 . . . . 5  |-  ( x  =  y  ->  ( S `  x )  =  ( S `  y ) )
1816, 17eleq12d 2364 . . . 4  |-  ( x  =  y  ->  (
( ( x  e.  I  |->  A ) `  x )  e.  ( S `  x )  <-> 
( ( x  e.  I  |->  A ) `  y )  e.  ( S `  y ) ) )
1911, 15, 18cbvral 2773 . . 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 ) )
2010, 19sylib 188 . 2  |-  ( ph  ->  A. y  e.  I 
( ( x  e.  I  |->  A ) `  y )  e.  ( S `  y ) )
21 dprdwd.4 . 2  |-  ( ph  ->  ( `' ( x  e.  I  |->  A )
" ( _V  \  {  .0.  } ) )  e.  Fin )
22 dprdff.w . . 3  |-  W  =  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V  \  {  .0.  } ) )  e. 
Fin }
23 dprdff.1 . . 3  |-  ( ph  ->  G dom DProd  S )
24 dprdff.2 . . 3  |-  ( ph  ->  dom  S  =  I )
2522, 23, 24dprdw 15261 . 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 ) ) )
265, 20, 21, 25mpbir3and 1135 1  |-  ( ph  ->  ( x  e.  I  |->  A )  e.  W
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   {crab 2560   _Vcvv 2801    \ cdif 3162   {csn 3653   class class class wbr 4039    e. cmpt 4093   `'ccnv 4704   dom cdm 4705   "cima 4708    Fn wfn 5266   ` cfv 5271   X_cixp 6833   Fincfn 6879   DProd cdprd 15247
This theorem is referenced by:  dprdfid  15268  dprdfinv  15270  dprdfadd  15271  dmdprdsplitlem  15288  dpjidcl  15309  dchrptlem3  20521
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-oprab 5878  df-mpt2 5879  df-ixp 6834  df-dprd 15249
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