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Theorem dprdf 17650
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 17641 . . . . . 6  |-  Rel  dom DProd
21brrelex2i 4879 . . . . 5  |-  ( G dom DProd  S  ->  S  e. 
_V )
3 dmexg 6729 . . . . 5  |-  ( S  e.  _V  ->  dom  S  e.  _V )
42, 3syl 17 . . . 4  |-  ( G dom DProd  S  ->  dom  S  e.  _V )
5 eqid 2453 . . . 4  |-  dom  S  =  dom  S
6 eqid 2453 . . . . 5  |-  (Cntz `  G )  =  (Cntz `  G )
7 eqid 2453 . . . . 5  |-  ( 0g
`  G )  =  ( 0g `  G
)
8 eqid 2453 . . . . 5  |-  (mrCls `  (SubGrp `  G ) )  =  (mrCls `  (SubGrp `  G ) )
96, 7, 8dmdprd 17642 . . . 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 669 . . 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 245 . 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 1022 1  |-  ( G dom DProd  S  ->  S : dom  S --> (SubGrp `  G )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 986    = wceq 1446    e. wcel 1889   A.wral 2739   _Vcvv 3047    \ cdif 3403    i^i cin 3405    C_ wss 3406   {csn 3970   U.cuni 4201   class class class wbr 4405   dom cdm 4837   "cima 4840   -->wf 5581   ` cfv 5585   0gc0g 15350  mrClscmrc 15501   Grpcgrp 16681  SubGrpcsubg 16823  Cntzccntz 16981   DProd cdprd 17637
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-reu 2746  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-op 3977  df-uni 4202  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-id 4752  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-oprab 6299  df-mpt2 6300  df-1st 6798  df-2nd 6799  df-ixp 7528  df-dprd 17639
This theorem is referenced by:  dprdf2  17651  dprdsubg  17669  dprdspan  17672  subgdprd  17680  ablfaclem2  17731  ablfac2  17734
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