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Theorem dprdff 15570
Description: A finitely supported function in  S is a function into the base. (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 )
dprdff.b  |-  B  =  ( Base `  G
)
Assertion
Ref Expression
dprdff  |-  ( ph  ->  F : I --> B )
Distinct variable groups:    h, F    h, i, I    .0. , h    S, h, i
Allowed substitution hints:    ph( h, i)    B( h, i)    F( i)    G( h, i)    W( h, i)    .0. ( i)

Proof of Theorem dprdff
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 dprdff.3 . . . 4  |-  ( ph  ->  F  e.  W )
2 dprdff.w . . . . 5  |-  W  =  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V  \  {  .0.  } ) )  e. 
Fin }
3 dprdff.1 . . . . 5  |-  ( ph  ->  G dom DProd  S )
4 dprdff.2 . . . . 5  |-  ( ph  ->  dom  S  =  I )
52, 3, 4dprdw 15568 . . . 4  |-  ( ph  ->  ( F  e.  W  <->  ( F  Fn  I  /\  A. x  e.  I  ( F `  x )  e.  ( S `  x )  /\  ( `' F " ( _V 
\  {  .0.  }
) )  e.  Fin ) ) )
61, 5mpbid 202 . . 3  |-  ( ph  ->  ( F  Fn  I  /\  A. x  e.  I 
( F `  x
)  e.  ( S `
 x )  /\  ( `' F " ( _V 
\  {  .0.  }
) )  e.  Fin ) )
76simp1d 969 . 2  |-  ( ph  ->  F  Fn  I )
86simp2d 970 . . 3  |-  ( ph  ->  A. x  e.  I 
( F `  x
)  e.  ( S `
 x ) )
93, 4dprdf2 15565 . . . . . . 7  |-  ( ph  ->  S : I --> (SubGrp `  G ) )
109ffvelrnda 5870 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  ( S `  x )  e.  (SubGrp `  G )
)
11 dprdff.b . . . . . . 7  |-  B  =  ( Base `  G
)
1211subgss 14945 . . . . . 6  |-  ( ( S `  x )  e.  (SubGrp `  G
)  ->  ( S `  x )  C_  B
)
1310, 12syl 16 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  ( S `  x )  C_  B )
1413sseld 3347 . . . 4  |-  ( (
ph  /\  x  e.  I )  ->  (
( F `  x
)  e.  ( S `
 x )  -> 
( F `  x
)  e.  B ) )
1514ralimdva 2784 . . 3  |-  ( ph  ->  ( A. x  e.  I  ( F `  x )  e.  ( S `  x )  ->  A. x  e.  I 
( F `  x
)  e.  B ) )
168, 15mpd 15 . 2  |-  ( ph  ->  A. x  e.  I 
( F `  x
)  e.  B )
17 ffnfv 5894 . 2  |-  ( F : I --> B  <->  ( F  Fn  I  /\  A. x  e.  I  ( F `  x )  e.  B
) )
187, 16, 17sylanbrc 646 1  |-  ( ph  ->  F : I --> B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2705   {crab 2709   _Vcvv 2956    \ cdif 3317    C_ wss 3320   {csn 3814   class class class wbr 4212   `'ccnv 4877   dom cdm 4878   "cima 4881    Fn wfn 5449   -->wf 5450   ` cfv 5454   X_cixp 7063   Fincfn 7109   Basecbs 13469  SubGrpcsubg 14938   DProd cdprd 15554
This theorem is referenced by:  dprdfcntz  15573  dprdssv  15574  dprdfid  15575  dprdfinv  15577  dprdfadd  15578  dprdfsub  15579  dprdfeq0  15580  dprdf11  15581  dprdlub  15584  dmdprdsplitlem  15595  dprddisj2  15597  dpjidcl  15616
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-ixp 7064  df-subg 14941  df-dprd 15556
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