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Theorem dprdf 17257
Description: The function  S is a family of subgroups. (Contributed by Mario Carneiro, 25-Apr-2016.)
Assertion
Ref Expression
dprdf  |-  ( G dom DProd  S  ->  S : dom  S --> (SubGrp `  G )
)

Proof of Theorem dprdf
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 reldmdprd 17246 . . . . . 6  |-  Rel  dom DProd
21brrelex2i 5050 . . . . 5  |-  ( G dom DProd  S  ->  S  e. 
_V )
3 dmexg 6730 . . . . 5  |-  ( S  e.  _V  ->  dom  S  e.  _V )
42, 3syl 16 . . . 4  |-  ( G dom DProd  S  ->  dom  S  e.  _V )
5 eqid 2457 . . . 4  |-  dom  S  =  dom  S
6 eqid 2457 . . . . 5  |-  (Cntz `  G )  =  (Cntz `  G )
7 eqid 2457 . . . . 5  |-  ( 0g
`  G )  =  ( 0g `  G
)
8 eqid 2457 . . . . 5  |-  (mrCls `  (SubGrp `  G ) )  =  (mrCls `  (SubGrp `  G ) )
96, 7, 8dmdprd 17247 . . . 4  |-  ( ( dom  S  e.  _V  /\ 
dom  S  =  dom  S )  ->  ( G dom DProd  S  <->  ( G  e. 
Grp  /\  S : dom  S --> (SubGrp `  G )  /\  A. x  e.  dom  S ( A. y  e.  ( dom  S  \  { x } ) ( S `  x
)  C_  ( (Cntz `  G ) `  ( S `  y )
)  /\  ( ( S `  x )  i^i  ( (mrCls `  (SubGrp `  G ) ) `  U. ( S " ( dom  S  \  { x } ) ) ) )  =  { ( 0g `  G ) } ) ) ) )
104, 5, 9sylancl 662 . . 3  |-  ( G dom DProd  S  ->  ( G dom DProd  S  <->  ( G  e. 
Grp  /\  S : dom  S --> (SubGrp `  G )  /\  A. x  e.  dom  S ( A. y  e.  ( dom  S  \  { x } ) ( S `  x
)  C_  ( (Cntz `  G ) `  ( S `  y )
)  /\  ( ( S `  x )  i^i  ( (mrCls `  (SubGrp `  G ) ) `  U. ( S " ( dom  S  \  { x } ) ) ) )  =  { ( 0g `  G ) } ) ) ) )
1110ibi 241 . 2  |-  ( G dom DProd  S  ->  ( G  e.  Grp  /\  S : dom  S --> (SubGrp `  G )  /\  A. x  e.  dom  S ( A. y  e.  ( dom  S  \  {
x } ) ( S `  x ) 
C_  ( (Cntz `  G ) `  ( S `  y )
)  /\  ( ( S `  x )  i^i  ( (mrCls `  (SubGrp `  G ) ) `  U. ( S " ( dom  S  \  { x } ) ) ) )  =  { ( 0g `  G ) } ) ) )
1211simp2d 1009 1  |-  ( G dom DProd  S  ->  S : dom  S --> (SubGrp `  G )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819   A.wral 2807   _Vcvv 3109    \ cdif 3468    i^i cin 3470    C_ wss 3471   {csn 4032   U.cuni 4251   class class class wbr 4456   dom cdm 5008   "cima 5011   -->wf 5590   ` cfv 5594   0gc0g 14948  mrClscmrc 15091   Grpcgrp 16271  SubGrpcsubg 16413  Cntzccntz 16571   DProd cdprd 17242
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-ixp 7489  df-dprd 17244
This theorem is referenced by:  dprdf2  17258  dprdsubg  17289  dprdspan  17292  subgdprd  17300  ablfaclem2  17355  ablfac2  17358
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