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Theorem issubgo 25431
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 3715 . . . . . . 7  |-  ( GrpOp  i^i 
~P G )  C_  ~P G
2 pwexg 4640 . . . . . . 7  |-  ( G  e.  GrpOp  ->  ~P G  e.  _V )
3 ssexg 4602 . . . . . . 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 4018 . . . . . . . 8  |-  ( g  =  G  ->  ~P g  =  ~P G
)
65ineq2d 3696 . . . . . . 7  |-  ( g  =  G  ->  ( GrpOp  i^i  ~P g )  =  ( GrpOp  i^i  ~P G ) )
7 df-subgo 25430 . . . . . . 7  |-  SubGrpOp  =  ( g  e.  GrpOp  |->  ( GrpOp  i^i 
~P g ) )
86, 7fvmptg 5954 . . . . . 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 2527 . . . 4  |-  ( G  e.  GrpOp  ->  ( H  e.  ( SubGrpOp `  G )  <->  H  e.  ( GrpOp  i^i  ~P G ) ) )
11 elin 3683 . . . . 5  |-  ( H  e.  ( GrpOp  i^i  ~P G )  <->  ( H  e.  GrpOp  /\  H  e.  ~P G ) )
12 elpwg 4023 . . . . . 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 5509 . . . 4  |-  dom  SubGrpOp  C_  GrpOp
18 elfvdm 5898 . . . 4  |-  ( H  e.  ( SubGrpOp `  G
)  ->  G  e.  dom 
SubGrpOp )
1917, 18sseldi 3497 . . 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 977 . 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 973    = wceq 1395    e. wcel 1819   _Vcvv 3109    i^i cin 3470    C_ wss 3471   ~Pcpw 4015   dom cdm 5008   ` cfv 5594   GrpOpcgr 25314   SubGrpOpcsubgo 25429
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fv 5602  df-subgo 25430
This theorem is referenced by:  subgores  25432  subgoid  25435  subgoinv  25436  issubgoi  25438  subgoablo  25439  ghsubgolemOLD  25498  hhssabloi  26304  ghomfo  29206  ghomgsg  29208
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