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Theorem dprdff 15529
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 15527 . . . 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 15524 . . . . . . 7  |-  ( ph  ->  S : I --> (SubGrp `  G ) )
109ffvelrnda 5833 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  ( S `  x )  e.  (SubGrp `  G )
)
11 dprdff.b . . . . . . 7  |-  B  =  ( Base `  G
)
1211subgss 14904 . . . . . 6  |-  ( ( S `  x )  e.  (SubGrp `  G
)  ->  ( S `  x )  C_  B
)
1310, 12syl 16 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  ( S `  x )  C_  B )
1413sseld 3311 . . . 4  |-  ( (
ph  /\  x  e.  I )  ->  (
( F `  x
)  e.  ( S `
 x )  -> 
( F `  x
)  e.  B ) )
1514ralimdva 2748 . . 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 5857 . 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 1649    e. wcel 1721   A.wral 2670   {crab 2674   _Vcvv 2920    \ cdif 3281    C_ wss 3284   {csn 3778   class class class wbr 4176   `'ccnv 4840   dom cdm 4841   "cima 4844    Fn wfn 5412   -->wf 5413   ` cfv 5417   X_cixp 7026   Fincfn 7072   Basecbs 13428  SubGrpcsubg 14897   DProd cdprd 15513
This theorem is referenced by:  dprdfcntz  15532  dprdssv  15533  dprdfid  15534  dprdfinv  15536  dprdfadd  15537  dprdfsub  15538  dprdfeq0  15539  dprdf11  15540  dprdlub  15543  dmdprdsplitlem  15554  dprddisj2  15556  dpjidcl  15575
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-reu 2677  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1st 6312  df-2nd 6313  df-ixp 7027  df-subg 14900  df-dprd 15515
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