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Theorem issubg 14871
Description: The subgroup predicate. (Contributed by Mario Carneiro, 2-Dec-2014.)
Hypothesis
Ref Expression
issubg.b  |-  B  =  ( Base `  G
)
Assertion
Ref Expression
issubg  |-  ( S  e.  (SubGrp `  G
)  <->  ( G  e. 
Grp  /\  S  C_  B  /\  ( Gs  S )  e.  Grp ) )

Proof of Theorem issubg
Dummy variables  w  s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-subg 14868 . . . 4  |- SubGrp  =  ( w  e.  Grp  |->  { s  e.  ~P ( Base `  w )  |  ( ws  s )  e. 
Grp } )
21dmmptss 5306 . . 3  |-  dom SubGrp  C_  Grp
3 elfvdm 5697 . . 3  |-  ( S  e.  (SubGrp `  G
)  ->  G  e.  dom SubGrp )
42, 3sseldi 3289 . 2  |-  ( S  e.  (SubGrp `  G
)  ->  G  e.  Grp )
5 simp1 957 . 2  |-  ( ( G  e.  Grp  /\  S  C_  B  /\  ( Gs  S )  e.  Grp )  ->  G  e.  Grp )
6 fveq2 5668 . . . . . . . . . 10  |-  ( w  =  G  ->  ( Base `  w )  =  ( Base `  G
) )
7 issubg.b . . . . . . . . . 10  |-  B  =  ( Base `  G
)
86, 7syl6eqr 2437 . . . . . . . . 9  |-  ( w  =  G  ->  ( Base `  w )  =  B )
98pweqd 3747 . . . . . . . 8  |-  ( w  =  G  ->  ~P ( Base `  w )  =  ~P B )
10 oveq1 6027 . . . . . . . . 9  |-  ( w  =  G  ->  (
ws  s )  =  ( Gs  s ) )
1110eleq1d 2453 . . . . . . . 8  |-  ( w  =  G  ->  (
( ws  s )  e. 
Grp 
<->  ( Gs  s )  e. 
Grp ) )
129, 11rabeqbidv 2894 . . . . . . 7  |-  ( w  =  G  ->  { s  e.  ~P ( Base `  w )  |  ( ws  s )  e.  Grp }  =  { s  e. 
~P B  |  ( Gs  s )  e.  Grp } )
13 fvex 5682 . . . . . . . . . 10  |-  ( Base `  G )  e.  _V
147, 13eqeltri 2457 . . . . . . . . 9  |-  B  e. 
_V
1514pwex 4323 . . . . . . . 8  |-  ~P B  e.  _V
1615rabex 4295 . . . . . . 7  |-  { s  e.  ~P B  | 
( Gs  s )  e. 
Grp }  e.  _V
1712, 1, 16fvmpt 5745 . . . . . 6  |-  ( G  e.  Grp  ->  (SubGrp `  G )  =  {
s  e.  ~P B  |  ( Gs  s )  e.  Grp } )
1817eleq2d 2454 . . . . 5  |-  ( G  e.  Grp  ->  ( S  e.  (SubGrp `  G
)  <->  S  e.  { s  e.  ~P B  | 
( Gs  s )  e. 
Grp } ) )
19 oveq2 6028 . . . . . . . 8  |-  ( s  =  S  ->  ( Gs  s )  =  ( Gs  S ) )
2019eleq1d 2453 . . . . . . 7  |-  ( s  =  S  ->  (
( Gs  s )  e. 
Grp 
<->  ( Gs  S )  e.  Grp ) )
2120elrab 3035 . . . . . 6  |-  ( S  e.  { s  e. 
~P B  |  ( Gs  s )  e.  Grp }  <-> 
( S  e.  ~P B  /\  ( Gs  S )  e.  Grp ) )
2214elpw2 4305 . . . . . . 7  |-  ( S  e.  ~P B  <->  S  C_  B
)
2322anbi1i 677 . . . . . 6  |-  ( ( S  e.  ~P B  /\  ( Gs  S )  e.  Grp ) 
<->  ( S  C_  B  /\  ( Gs  S )  e.  Grp ) )
2421, 23bitri 241 . . . . 5  |-  ( S  e.  { s  e. 
~P B  |  ( Gs  s )  e.  Grp }  <-> 
( S  C_  B  /\  ( Gs  S )  e.  Grp ) )
2518, 24syl6bb 253 . . . 4  |-  ( G  e.  Grp  ->  ( S  e.  (SubGrp `  G
)  <->  ( S  C_  B  /\  ( Gs  S )  e.  Grp ) ) )
26 ibar 491 . . . 4  |-  ( G  e.  Grp  ->  (
( S  C_  B  /\  ( Gs  S )  e.  Grp ) 
<->  ( G  e.  Grp  /\  ( S  C_  B  /\  ( Gs  S )  e.  Grp ) ) ) )
2725, 26bitrd 245 . . 3  |-  ( G  e.  Grp  ->  ( S  e.  (SubGrp `  G
)  <->  ( G  e. 
Grp  /\  ( S  C_  B  /\  ( Gs  S )  e.  Grp )
) ) )
28 3anass 940 . . 3  |-  ( ( G  e.  Grp  /\  S  C_  B  /\  ( Gs  S )  e.  Grp ) 
<->  ( G  e.  Grp  /\  ( S  C_  B  /\  ( Gs  S )  e.  Grp ) ) )
2927, 28syl6bbr 255 . 2  |-  ( G  e.  Grp  ->  ( S  e.  (SubGrp `  G
)  <->  ( G  e. 
Grp  /\  S  C_  B  /\  ( Gs  S )  e.  Grp ) ) )
304, 5, 29pm5.21nii 343 1  |-  ( S  e.  (SubGrp `  G
)  <->  ( G  e. 
Grp  /\  S  C_  B  /\  ( Gs  S )  e.  Grp ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   {crab 2653   _Vcvv 2899    C_ wss 3263   ~Pcpw 3742   dom cdm 4818   ` cfv 5394  (class class class)co 6020   Basecbs 13396   ↾s cress 13397   Grpcgrp 14612  SubGrpcsubg 14865
This theorem is referenced by:  subgss  14872  subgid  14873  subggrp  14874  subgrcl  14876  issubg2  14886  subsubg  14890  opprsubg  15668  subrgsubg  15801  cphsubrglem  19011  subofld  24071
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fv 5402  df-ov 6023  df-subg 14868
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