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Theorem issubgo 23802
Description: The predicate "is a subgroup of  G." (Contributed by Paul Chapman, 3-Mar-2008.) (Revised by Mario Carneiro, 12-Jul-2014.) (New usage is discouraged.)
Assertion
Ref Expression
issubgo  |-  ( H  e.  ( SubGrpOp `  G
)  <->  ( G  e. 
GrpOp  /\  H  e.  GrpOp  /\  H  C_  G )
)

Proof of Theorem issubgo
Dummy variable  g is distinct from all other variables.
StepHypRef Expression
1 inss2 3583 . . . . . . 7  |-  ( GrpOp  i^i 
~P G )  C_  ~P G
2 pwexg 4488 . . . . . . 7  |-  ( G  e.  GrpOp  ->  ~P G  e.  _V )
3 ssexg 4450 . . . . . . 7  |-  ( ( ( GrpOp  i^i  ~P G
)  C_  ~P G  /\  ~P G  e.  _V )  ->  ( GrpOp  i^i  ~P G )  e.  _V )
41, 2, 3sylancr 663 . . . . . 6  |-  ( G  e.  GrpOp  ->  ( GrpOp  i^i 
~P G )  e. 
_V )
5 pweq 3875 . . . . . . . 8  |-  ( g  =  G  ->  ~P g  =  ~P G
)
65ineq2d 3564 . . . . . . 7  |-  ( g  =  G  ->  ( GrpOp  i^i  ~P g )  =  ( GrpOp  i^i  ~P G ) )
7 df-subgo 23801 . . . . . . 7  |-  SubGrpOp  =  ( g  e.  GrpOp  |->  ( GrpOp  i^i 
~P g ) )
86, 7fvmptg 5784 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  ( GrpOp  i^i  ~P G )  e.  _V )  -> 
( SubGrpOp `  G )  =  ( GrpOp  i^i  ~P G ) )
94, 8mpdan 668 . . . . 5  |-  ( G  e.  GrpOp  ->  ( SubGrpOp `  G )  =  (
GrpOp  i^i  ~P G ) )
109eleq2d 2510 . . . 4  |-  ( G  e.  GrpOp  ->  ( H  e.  ( SubGrpOp `  G )  <->  H  e.  ( GrpOp  i^i  ~P G ) ) )
11 elin 3551 . . . . 5  |-  ( H  e.  ( GrpOp  i^i  ~P G )  <->  ( H  e.  GrpOp  /\  H  e.  ~P G ) )
12 elpwg 3880 . . . . . 6  |-  ( H  e.  GrpOp  ->  ( H  e.  ~P G  <->  H  C_  G
) )
1312pm5.32i 637 . . . . 5  |-  ( ( H  e.  GrpOp  /\  H  e.  ~P G )  <->  ( H  e.  GrpOp  /\  H  C_  G
) )
1411, 13bitri 249 . . . 4  |-  ( H  e.  ( GrpOp  i^i  ~P G )  <->  ( H  e.  GrpOp  /\  H  C_  G
) )
1510, 14syl6bb 261 . . 3  |-  ( G  e.  GrpOp  ->  ( H  e.  ( SubGrpOp `  G )  <->  ( H  e.  GrpOp  /\  H  C_  G ) ) )
1615pm5.32i 637 . 2  |-  ( ( G  e.  GrpOp  /\  H  e.  ( SubGrpOp `  G )
)  <->  ( G  e. 
GrpOp  /\  ( H  e. 
GrpOp  /\  H  C_  G
) ) )
177dmmptss 5346 . . . 4  |-  dom  SubGrpOp  C_  GrpOp
18 elfvdm 5728 . . . 4  |-  ( H  e.  ( SubGrpOp `  G
)  ->  G  e.  dom 
SubGrpOp )
1917, 18sseldi 3366 . . 3  |-  ( H  e.  ( SubGrpOp `  G
)  ->  G  e.  GrpOp
)
2019pm4.71ri 633 . 2  |-  ( H  e.  ( SubGrpOp `  G
)  <->  ( G  e. 
GrpOp  /\  H  e.  (
SubGrpOp `  G ) ) )
21 3anass 969 . 2  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  H  C_  G
)  <->  ( G  e. 
GrpOp  /\  ( H  e. 
GrpOp  /\  H  C_  G
) ) )
2216, 20, 213bitr4i 277 1  |-  ( H  e.  ( SubGrpOp `  G
)  <->  ( G  e. 
GrpOp  /\  H  e.  GrpOp  /\  H  C_  G )
)
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   _Vcvv 2984    i^i cin 3339    C_ wss 3340   ~Pcpw 3872   dom cdm 4852   ` cfv 5430   GrpOpcgr 23685   SubGrpOpcsubgo 23800
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-rab 2736  df-v 2986  df-sbc 3199  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-br 4305  df-opab 4363  df-mpt 4364  df-id 4648  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fv 5438  df-subgo 23801
This theorem is referenced by:  subgores  23803  subgoid  23806  subgoinv  23807  issubgoi  23809  subgoablo  23810  ghsubgolem  23869  hhssabloi  24675  ghomfo  27322  ghomgsg  27324
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