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Theorem dprdf 15257
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 15251 . . . . . 6  |-  Rel  dom DProd
21brrelex2i 4746 . . . . 5  |-  ( G dom DProd  S  ->  S  e. 
_V )
3 dmexg 4955 . . . . 5  |-  ( S  e.  _V  ->  dom  S  e.  _V )
42, 3syl 15 . . . 4  |-  ( G dom DProd  S  ->  dom  S  e.  _V )
5 eqid 2296 . . . 4  |-  dom  S  =  dom  S
6 eqid 2296 . . . . 5  |-  (Cntz `  G )  =  (Cntz `  G )
7 eqid 2296 . . . . 5  |-  ( 0g
`  G )  =  ( 0g `  G
)
8 eqid 2296 . . . . 5  |-  (mrCls `  (SubGrp `  G ) )  =  (mrCls `  (SubGrp `  G ) )
96, 7, 8dmdprd 15252 . . . 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 643 . . 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 232 . 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 968 1  |-  ( G dom DProd  S  ->  S : dom  S --> (SubGrp `  G )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556   _Vcvv 2801    \ cdif 3162    i^i cin 3164    C_ wss 3165   {csn 3653   U.cuni 3843   class class class wbr 4039   dom cdm 4705   "cima 4708   -->wf 5267   ` cfv 5271   0gc0g 13416  mrClscmrc 13501   Grpcgrp 14378  SubGrpcsubg 14631  Cntzccntz 14807   DProd cdprd 15247
This theorem is referenced by:  dprdf2  15258  dprdsubg  15275  dprdspan  15278  subgdprd  15286  ablfaclem2  15337  ablfac2  15340
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-ixp 6834  df-dprd 15249
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