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Theorem issubg 14637
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 14634 . . . 4  |- SubGrp  =  ( w  e.  Grp  |->  { s  e.  ~P ( Base `  w )  |  ( ws  s )  e. 
Grp } )
21dmmptss 5185 . . 3  |-  dom SubGrp  C_  Grp
3 elfvdm 5570 . . 3  |-  ( S  e.  (SubGrp `  G
)  ->  G  e.  dom SubGrp )
42, 3sseldi 3191 . 2  |-  ( S  e.  (SubGrp `  G
)  ->  G  e.  Grp )
5 simp1 955 . 2  |-  ( ( G  e.  Grp  /\  S  C_  B  /\  ( Gs  S )  e.  Grp )  ->  G  e.  Grp )
6 fveq2 5541 . . . . . . . . . 10  |-  ( w  =  G  ->  ( Base `  w )  =  ( Base `  G
) )
7 issubg.b . . . . . . . . . 10  |-  B  =  ( Base `  G
)
86, 7syl6eqr 2346 . . . . . . . . 9  |-  ( w  =  G  ->  ( Base `  w )  =  B )
98pweqd 3643 . . . . . . . 8  |-  ( w  =  G  ->  ~P ( Base `  w )  =  ~P B )
10 oveq1 5881 . . . . . . . . 9  |-  ( w  =  G  ->  (
ws  s )  =  ( Gs  s ) )
1110eleq1d 2362 . . . . . . . 8  |-  ( w  =  G  ->  (
( ws  s )  e. 
Grp 
<->  ( Gs  s )  e. 
Grp ) )
129, 11rabeqbidv 2796 . . . . . . 7  |-  ( w  =  G  ->  { s  e.  ~P ( Base `  w )  |  ( ws  s )  e.  Grp }  =  { s  e. 
~P B  |  ( Gs  s )  e.  Grp } )
13 fvex 5555 . . . . . . . . . 10  |-  ( Base `  G )  e.  _V
147, 13eqeltri 2366 . . . . . . . . 9  |-  B  e. 
_V
1514pwex 4209 . . . . . . . 8  |-  ~P B  e.  _V
1615rabex 4181 . . . . . . 7  |-  { s  e.  ~P B  | 
( Gs  s )  e. 
Grp }  e.  _V
1712, 1, 16fvmpt 5618 . . . . . 6  |-  ( G  e.  Grp  ->  (SubGrp `  G )  =  {
s  e.  ~P B  |  ( Gs  s )  e.  Grp } )
1817eleq2d 2363 . . . . 5  |-  ( G  e.  Grp  ->  ( S  e.  (SubGrp `  G
)  <->  S  e.  { s  e.  ~P B  | 
( Gs  s )  e. 
Grp } ) )
19 oveq2 5882 . . . . . . . 8  |-  ( s  =  S  ->  ( Gs  s )  =  ( Gs  S ) )
2019eleq1d 2362 . . . . . . 7  |-  ( s  =  S  ->  (
( Gs  s )  e. 
Grp 
<->  ( Gs  S )  e.  Grp ) )
2120elrab 2936 . . . . . 6  |-  ( S  e.  { s  e. 
~P B  |  ( Gs  s )  e.  Grp }  <-> 
( S  e.  ~P B  /\  ( Gs  S )  e.  Grp ) )
2214elpw2 4191 . . . . . . 7  |-  ( S  e.  ~P B  <->  S  C_  B
)
2322anbi1i 676 . . . . . 6  |-  ( ( S  e.  ~P B  /\  ( Gs  S )  e.  Grp ) 
<->  ( S  C_  B  /\  ( Gs  S )  e.  Grp ) )
2421, 23bitri 240 . . . . 5  |-  ( S  e.  { s  e. 
~P B  |  ( Gs  s )  e.  Grp }  <-> 
( S  C_  B  /\  ( Gs  S )  e.  Grp ) )
2518, 24syl6bb 252 . . . 4  |-  ( G  e.  Grp  ->  ( S  e.  (SubGrp `  G
)  <->  ( S  C_  B  /\  ( Gs  S )  e.  Grp ) ) )
26 ibar 490 . . . 4  |-  ( G  e.  Grp  ->  (
( S  C_  B  /\  ( Gs  S )  e.  Grp ) 
<->  ( G  e.  Grp  /\  ( S  C_  B  /\  ( Gs  S )  e.  Grp ) ) ) )
2725, 26bitrd 244 . . 3  |-  ( G  e.  Grp  ->  ( S  e.  (SubGrp `  G
)  <->  ( G  e. 
Grp  /\  ( S  C_  B  /\  ( Gs  S )  e.  Grp )
) ) )
28 3anass 938 . . 3  |-  ( ( G  e.  Grp  /\  S  C_  B  /\  ( Gs  S )  e.  Grp ) 
<->  ( G  e.  Grp  /\  ( S  C_  B  /\  ( Gs  S )  e.  Grp ) ) )
2927, 28syl6bbr 254 . 2  |-  ( G  e.  Grp  ->  ( S  e.  (SubGrp `  G
)  <->  ( G  e. 
Grp  /\  S  C_  B  /\  ( Gs  S )  e.  Grp ) ) )
304, 5, 29pm5.21nii 342 1  |-  ( S  e.  (SubGrp `  G
)  <->  ( G  e. 
Grp  /\  S  C_  B  /\  ( Gs  S )  e.  Grp ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   {crab 2560   _Vcvv 2801    C_ wss 3165   ~Pcpw 3638   dom cdm 4705   ` cfv 5271  (class class class)co 5874   Basecbs 13164   ↾s cress 13165   Grpcgrp 14378  SubGrpcsubg 14631
This theorem is referenced by:  subgss  14638  subgid  14639  subggrp  14640  subgrcl  14642  issubg2  14652  subsubg  14656  opprsubg  15434  subrgsubg  15567  cphsubrglem  18629
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-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
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-rab 2565  df-v 2803  df-sbc 3005  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-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-fv 5279  df-ov 5877  df-subg 14634
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