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Theorem issubg 16072
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 16069 . . . 4  |- SubGrp  =  ( w  e.  Grp  |->  { s  e.  ~P ( Base `  w )  |  ( ws  s )  e. 
Grp } )
21dmmptss 5509 . . 3  |-  dom SubGrp  C_  Grp
3 elfvdm 5898 . . 3  |-  ( S  e.  (SubGrp `  G
)  ->  G  e.  dom SubGrp )
42, 3sseldi 3507 . 2  |-  ( S  e.  (SubGrp `  G
)  ->  G  e.  Grp )
5 simp1 996 . 2  |-  ( ( G  e.  Grp  /\  S  C_  B  /\  ( Gs  S )  e.  Grp )  ->  G  e.  Grp )
6 fveq2 5872 . . . . . . . . . 10  |-  ( w  =  G  ->  ( Base `  w )  =  ( Base `  G
) )
7 issubg.b . . . . . . . . . 10  |-  B  =  ( Base `  G
)
86, 7syl6eqr 2526 . . . . . . . . 9  |-  ( w  =  G  ->  ( Base `  w )  =  B )
98pweqd 4021 . . . . . . . 8  |-  ( w  =  G  ->  ~P ( Base `  w )  =  ~P B )
10 oveq1 6302 . . . . . . . . 9  |-  ( w  =  G  ->  (
ws  s )  =  ( Gs  s ) )
1110eleq1d 2536 . . . . . . . 8  |-  ( w  =  G  ->  (
( ws  s )  e. 
Grp 
<->  ( Gs  s )  e. 
Grp ) )
129, 11rabeqbidv 3113 . . . . . . 7  |-  ( w  =  G  ->  { s  e.  ~P ( Base `  w )  |  ( ws  s )  e.  Grp }  =  { s  e. 
~P B  |  ( Gs  s )  e.  Grp } )
13 fvex 5882 . . . . . . . . . 10  |-  ( Base `  G )  e.  _V
147, 13eqeltri 2551 . . . . . . . . 9  |-  B  e. 
_V
1514pwex 4636 . . . . . . . 8  |-  ~P B  e.  _V
1615rabex 4604 . . . . . . 7  |-  { s  e.  ~P B  | 
( Gs  s )  e. 
Grp }  e.  _V
1712, 1, 16fvmpt 5957 . . . . . 6  |-  ( G  e.  Grp  ->  (SubGrp `  G )  =  {
s  e.  ~P B  |  ( Gs  s )  e.  Grp } )
1817eleq2d 2537 . . . . 5  |-  ( G  e.  Grp  ->  ( S  e.  (SubGrp `  G
)  <->  S  e.  { s  e.  ~P B  | 
( Gs  s )  e. 
Grp } ) )
19 oveq2 6303 . . . . . . . 8  |-  ( s  =  S  ->  ( Gs  s )  =  ( Gs  S ) )
2019eleq1d 2536 . . . . . . 7  |-  ( s  =  S  ->  (
( Gs  s )  e. 
Grp 
<->  ( Gs  S )  e.  Grp ) )
2120elrab 3266 . . . . . 6  |-  ( S  e.  { s  e. 
~P B  |  ( Gs  s )  e.  Grp }  <-> 
( S  e.  ~P B  /\  ( Gs  S )  e.  Grp ) )
2214elpw2 4617 . . . . . . 7  |-  ( S  e.  ~P B  <->  S  C_  B
)
2322anbi1i 695 . . . . . 6  |-  ( ( S  e.  ~P B  /\  ( Gs  S )  e.  Grp ) 
<->  ( S  C_  B  /\  ( Gs  S )  e.  Grp ) )
2421, 23bitri 249 . . . . 5  |-  ( S  e.  { s  e. 
~P B  |  ( Gs  s )  e.  Grp }  <-> 
( S  C_  B  /\  ( Gs  S )  e.  Grp ) )
2518, 24syl6bb 261 . . . 4  |-  ( G  e.  Grp  ->  ( S  e.  (SubGrp `  G
)  <->  ( S  C_  B  /\  ( Gs  S )  e.  Grp ) ) )
26 ibar 504 . . . 4  |-  ( G  e.  Grp  ->  (
( S  C_  B  /\  ( Gs  S )  e.  Grp ) 
<->  ( G  e.  Grp  /\  ( S  C_  B  /\  ( Gs  S )  e.  Grp ) ) ) )
2725, 26bitrd 253 . . 3  |-  ( G  e.  Grp  ->  ( S  e.  (SubGrp `  G
)  <->  ( G  e. 
Grp  /\  ( S  C_  B  /\  ( Gs  S )  e.  Grp )
) ) )
28 3anass 977 . . 3  |-  ( ( G  e.  Grp  /\  S  C_  B  /\  ( Gs  S )  e.  Grp ) 
<->  ( G  e.  Grp  /\  ( S  C_  B  /\  ( Gs  S )  e.  Grp ) ) )
2927, 28syl6bbr 263 . 2  |-  ( G  e.  Grp  ->  ( S  e.  (SubGrp `  G
)  <->  ( G  e. 
Grp  /\  S  C_  B  /\  ( Gs  S )  e.  Grp ) ) )
304, 5, 29pm5.21nii 353 1  |-  ( S  e.  (SubGrp `  G
)  <->  ( G  e. 
Grp  /\  S  C_  B  /\  ( Gs  S )  e.  Grp ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   {crab 2821   _Vcvv 3118    C_ wss 3481   ~Pcpw 4016   dom cdm 5005   ` cfv 5594  (class class class)co 6295   Basecbs 14506   ↾s cress 14507   Grpcgrp 15924  SubGrpcsubg 16066
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fv 5602  df-ov 6298  df-subg 16069
This theorem is referenced by:  subgss  16073  subgid  16074  subggrp  16075  subgrcl  16077  issubg2  16087  resgrpisgrp  16093  subsubg  16095  pgrpsubgsymgbi  16303  opprsubg  17155  subrgsubg  17304  cphsubrglem  21490  suborng  27637
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