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Theorem dprdff 16603
Description: A finitely supported function in  S is a function into the base. (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 )
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 finSupp  .0.  }
3 dprdff.1 . . . . 5  |-  ( ph  ->  G dom DProd  S )
4 dprdff.2 . . . . 5  |-  ( ph  ->  dom  S  =  I )
52, 3, 4dprdw 16601 . . . 4  |-  ( ph  ->  ( F  e.  W  <->  ( F  Fn  I  /\  A. x  e.  I  ( F `  x )  e.  ( S `  x )  /\  F finSupp  .0.  ) ) )
61, 5mpbid 210 . . 3  |-  ( ph  ->  ( F  Fn  I  /\  A. x  e.  I 
( F `  x
)  e.  ( S `
 x )  /\  F finSupp  .0.  ) )
76simp1d 1000 . 2  |-  ( ph  ->  F  Fn  I )
86simp2d 1001 . . 3  |-  ( ph  ->  A. x  e.  I 
( F `  x
)  e.  ( S `
 x ) )
93, 4dprdf2 16598 . . . . . . 7  |-  ( ph  ->  S : I --> (SubGrp `  G ) )
109ffvelrnda 5944 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  ( S `  x )  e.  (SubGrp `  G )
)
11 dprdff.b . . . . . . 7  |-  B  =  ( Base `  G
)
1211subgss 15786 . . . . . 6  |-  ( ( S `  x )  e.  (SubGrp `  G
)  ->  ( S `  x )  C_  B
)
1310, 12syl 16 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  ( S `  x )  C_  B )
1413sseld 3455 . . . 4  |-  ( (
ph  /\  x  e.  I )  ->  (
( F `  x
)  e.  ( S `
 x )  -> 
( F `  x
)  e.  B ) )
1514ralimdva 2824 . . 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 5970 . 2  |-  ( F : I --> B  <->  ( F  Fn  I  /\  A. x  e.  I  ( F `  x )  e.  B
) )
187, 16, 17sylanbrc 664 1  |-  ( ph  ->  F : I --> B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   A.wral 2795   {crab 2799    C_ wss 3428   class class class wbr 4392   dom cdm 4940    Fn wfn 5513   -->wf 5514   ` cfv 5518   X_cixp 7365   finSupp cfsupp 7723   Basecbs 14278  SubGrpcsubg 15779   DProd cdprd 16582
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 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474
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 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-1st 6679  df-2nd 6680  df-ixp 7366  df-subg 15782  df-dprd 16584
This theorem is referenced by:  dprdfcntz  16606  dprdssv  16613  dprdfid  16614  dprdfinv  16616  dprdfadd  16617  dprdfsub  16618  dprdfeq0  16619  dprdf11  16620  dprdlub  16630  dmdprdsplitlem  16641  dprddisj2  16644  dpjidcl  16664
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