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Theorem dprdff 16829
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 16827 . . . 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 1003 . 2  |-  ( ph  ->  F  Fn  I )
86simp2d 1004 . . 3  |-  ( ph  ->  A. x  e.  I 
( F `  x
)  e.  ( S `
 x ) )
93, 4dprdf2 16824 . . . . . . 7  |-  ( ph  ->  S : I --> (SubGrp `  G ) )
109ffvelrnda 6012 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  ( S `  x )  e.  (SubGrp `  G )
)
11 dprdff.b . . . . . . 7  |-  B  =  ( Base `  G
)
1211subgss 15990 . . . . . 6  |-  ( ( S `  x )  e.  (SubGrp `  G
)  ->  ( S `  x )  C_  B
)
1310, 12syl 16 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  ( S `  x )  C_  B )
1413sseld 3496 . . . 4  |-  ( (
ph  /\  x  e.  I )  ->  (
( F `  x
)  e.  ( S `
 x )  -> 
( F `  x
)  e.  B ) )
1514ralimdva 2865 . . 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 6038 . 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 968    = wceq 1374    e. wcel 1762   A.wral 2807   {crab 2811    C_ wss 3469   class class class wbr 4440   dom cdm 4992    Fn wfn 5574   -->wf 5575   ` cfv 5579   X_cixp 7459   finSupp cfsupp 7818   Basecbs 14479  SubGrpcsubg 15983   DProd cdprd 16808
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-1st 6774  df-2nd 6775  df-ixp 7460  df-subg 15986  df-dprd 16810
This theorem is referenced by:  dprdfcntz  16832  dprdssv  16839  dprdfid  16840  dprdfinv  16842  dprdfadd  16843  dprdfsub  16844  dprdfeq0  16845  dprdf11  16846  dprdlub  16856  dmdprdsplitlem  16867  dprddisj2  16870  dpjidcl  16890
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