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Theorem dprdff 15346
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 15344 . . . 4  |-  ( ph  ->  ( F  e.  W  <->  ( F  Fn  I  /\  A. x  e.  I  ( F `  x )  e.  ( S `  x )  /\  ( `' F " ( _V 
\  {  .0.  }
) )  e.  Fin ) ) )
61, 5mpbid 201 . . 3  |-  ( ph  ->  ( F  Fn  I  /\  A. x  e.  I 
( F `  x
)  e.  ( S `
 x )  /\  ( `' F " ( _V 
\  {  .0.  }
) )  e.  Fin ) )
76simp1d 967 . 2  |-  ( ph  ->  F  Fn  I )
86simp2d 968 . . 3  |-  ( ph  ->  A. x  e.  I 
( F `  x
)  e.  ( S `
 x ) )
93, 4dprdf2 15341 . . . . . . 7  |-  ( ph  ->  S : I --> (SubGrp `  G ) )
10 ffvelrn 5746 . . . . . . 7  |-  ( ( S : I --> (SubGrp `  G )  /\  x  e.  I )  ->  ( S `  x )  e.  (SubGrp `  G )
)
119, 10sylan 457 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  ( S `  x )  e.  (SubGrp `  G )
)
12 dprdff.b . . . . . . 7  |-  B  =  ( Base `  G
)
1312subgss 14721 . . . . . 6  |-  ( ( S `  x )  e.  (SubGrp `  G
)  ->  ( S `  x )  C_  B
)
1411, 13syl 15 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  ( S `  x )  C_  B )
1514sseld 3255 . . . 4  |-  ( (
ph  /\  x  e.  I )  ->  (
( F `  x
)  e.  ( S `
 x )  -> 
( F `  x
)  e.  B ) )
1615ralimdva 2697 . . 3  |-  ( ph  ->  ( A. x  e.  I  ( F `  x )  e.  ( S `  x )  ->  A. x  e.  I 
( F `  x
)  e.  B ) )
178, 16mpd 14 . 2  |-  ( ph  ->  A. x  e.  I 
( F `  x
)  e.  B )
18 ffnfv 5768 . 2  |-  ( F : I --> B  <->  ( F  Fn  I  /\  A. x  e.  I  ( F `  x )  e.  B
) )
197, 17, 18sylanbrc 645 1  |-  ( ph  ->  F : I --> B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1642    e. wcel 1710   A.wral 2619   {crab 2623   _Vcvv 2864    \ cdif 3225    C_ wss 3228   {csn 3716   class class class wbr 4104   `'ccnv 4770   dom cdm 4771   "cima 4774    Fn wfn 5332   -->wf 5333   ` cfv 5337   X_cixp 6905   Fincfn 6951   Basecbs 13245  SubGrpcsubg 14714   DProd cdprd 15330
This theorem is referenced by:  dprdfcntz  15349  dprdssv  15350  dprdfid  15351  dprdfinv  15353  dprdfadd  15354  dprdfsub  15355  dprdfeq0  15356  dprdf11  15357  dprdlub  15360  dmdprdsplitlem  15371  dprddisj2  15373  dpjidcl  15392
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-1st 6209  df-2nd 6210  df-ixp 6906  df-subg 14717  df-dprd 15332
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