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Theorem dprdffi 15265
Description: The function  S is a family of subgroups. (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 )
dprdff.3  |-  ( ph  ->  F  e.  W )
Assertion
Ref Expression
dprdffi  |-  ( ph  ->  ( `' F "
( _V  \  {  .0.  } ) )  e. 
Fin )
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)    .0. ( i)

Proof of Theorem dprdffi
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 dprdff.3 . . 3  |-  ( ph  ->  F  e.  W )
2 dprdff.w . . . 4  |-  W  =  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V  \  {  .0.  } ) )  e. 
Fin }
3 dprdff.1 . . . 4  |-  ( ph  ->  G dom DProd  S )
4 dprdff.2 . . . 4  |-  ( ph  ->  dom  S  =  I )
52, 3, 4dprdw 15261 . . 3  |-  ( ph  ->  ( F  e.  W  <->  ( F  Fn  I  /\  A. x  e.  I  ( F `  x )  e.  ( S `  x )  /\  ( `' F " ( _V 
\  {  .0.  }
) )  e.  Fin ) ) )
61, 5mpbid 201 . 2  |-  ( ph  ->  ( F  Fn  I  /\  A. x  e.  I 
( F `  x
)  e.  ( S `
 x )  /\  ( `' F " ( _V 
\  {  .0.  }
) )  e.  Fin ) )
76simp3d 969 1  |-  ( ph  ->  ( `' F "
( _V  \  {  .0.  } ) )  e. 
Fin )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556   {crab 2560   _Vcvv 2801    \ cdif 3162   {csn 3653   class class class wbr 4039   `'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:  dprdssv  15267  dprdfinv  15270  dprdfadd  15271  dprdfeq0  15273  dprdlub  15277  dmdprdsplitlem  15288  dpjidcl  15309
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-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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