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Theorem issubg 14936
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 14933 . . . 4  |- SubGrp  =  ( w  e.  Grp  |->  { s  e.  ~P ( Base `  w )  |  ( ws  s )  e. 
Grp } )
21dmmptss 5358 . . 3  |-  dom SubGrp  C_  Grp
3 elfvdm 5749 . . 3  |-  ( S  e.  (SubGrp `  G
)  ->  G  e.  dom SubGrp )
42, 3sseldi 3338 . 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 5720 . . . . . . . . . 10  |-  ( w  =  G  ->  ( Base `  w )  =  ( Base `  G
) )
7 issubg.b . . . . . . . . . 10  |-  B  =  ( Base `  G
)
86, 7syl6eqr 2485 . . . . . . . . 9  |-  ( w  =  G  ->  ( Base `  w )  =  B )
98pweqd 3796 . . . . . . . 8  |-  ( w  =  G  ->  ~P ( Base `  w )  =  ~P B )
10 oveq1 6080 . . . . . . . . 9  |-  ( w  =  G  ->  (
ws  s )  =  ( Gs  s ) )
1110eleq1d 2501 . . . . . . . 8  |-  ( w  =  G  ->  (
( ws  s )  e. 
Grp 
<->  ( Gs  s )  e. 
Grp ) )
129, 11rabeqbidv 2943 . . . . . . 7  |-  ( w  =  G  ->  { s  e.  ~P ( Base `  w )  |  ( ws  s )  e.  Grp }  =  { s  e. 
~P B  |  ( Gs  s )  e.  Grp } )
13 fvex 5734 . . . . . . . . . 10  |-  ( Base `  G )  e.  _V
147, 13eqeltri 2505 . . . . . . . . 9  |-  B  e. 
_V
1514pwex 4374 . . . . . . . 8  |-  ~P B  e.  _V
1615rabex 4346 . . . . . . 7  |-  { s  e.  ~P B  | 
( Gs  s )  e. 
Grp }  e.  _V
1712, 1, 16fvmpt 5798 . . . . . 6  |-  ( G  e.  Grp  ->  (SubGrp `  G )  =  {
s  e.  ~P B  |  ( Gs  s )  e.  Grp } )
1817eleq2d 2502 . . . . 5  |-  ( G  e.  Grp  ->  ( S  e.  (SubGrp `  G
)  <->  S  e.  { s  e.  ~P B  | 
( Gs  s )  e. 
Grp } ) )
19 oveq2 6081 . . . . . . . 8  |-  ( s  =  S  ->  ( Gs  s )  =  ( Gs  S ) )
2019eleq1d 2501 . . . . . . 7  |-  ( s  =  S  ->  (
( Gs  s )  e. 
Grp 
<->  ( Gs  S )  e.  Grp ) )
2120elrab 3084 . . . . . 6  |-  ( S  e.  { s  e. 
~P B  |  ( Gs  s )  e.  Grp }  <-> 
( S  e.  ~P B  /\  ( Gs  S )  e.  Grp ) )
2214elpw2 4356 . . . . . . 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 1652    e. wcel 1725   {crab 2701   _Vcvv 2948    C_ wss 3312   ~Pcpw 3791   dom cdm 4870   ` cfv 5446  (class class class)co 6073   Basecbs 13461   ↾s cress 13462   Grpcgrp 14677  SubGrpcsubg 14930
This theorem is referenced by:  subgss  14937  subgid  14938  subggrp  14939  subgrcl  14941  issubg2  14951  subsubg  14955  opprsubg  15733  subrgsubg  15866  cphsubrglem  19132  subofld  24237
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fv 5454  df-ov 6076  df-subg 14933
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