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Theorem issubgo 25009
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 3719 . . . . . . 7  |-  ( GrpOp  i^i 
~P G )  C_  ~P G
2 pwexg 4631 . . . . . . 7  |-  ( G  e.  GrpOp  ->  ~P G  e.  _V )
3 ssexg 4593 . . . . . . 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 4013 . . . . . . . 8  |-  ( g  =  G  ->  ~P g  =  ~P G
)
65ineq2d 3700 . . . . . . 7  |-  ( g  =  G  ->  ( GrpOp  i^i  ~P g )  =  ( GrpOp  i^i  ~P G ) )
7 df-subgo 25008 . . . . . . 7  |-  SubGrpOp  =  ( g  e.  GrpOp  |->  ( GrpOp  i^i 
~P g ) )
86, 7fvmptg 5948 . . . . . 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 2537 . . . 4  |-  ( G  e.  GrpOp  ->  ( H  e.  ( SubGrpOp `  G )  <->  H  e.  ( GrpOp  i^i  ~P G ) ) )
11 elin 3687 . . . . 5  |-  ( H  e.  ( GrpOp  i^i  ~P G )  <->  ( H  e.  GrpOp  /\  H  e.  ~P G ) )
12 elpwg 4018 . . . . . 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 5503 . . . 4  |-  dom  SubGrpOp  C_  GrpOp
18 elfvdm 5892 . . . 4  |-  ( H  e.  ( SubGrpOp `  G
)  ->  G  e.  dom 
SubGrpOp )
1917, 18sseldi 3502 . . 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 1379    e. wcel 1767   _Vcvv 3113    i^i cin 3475    C_ wss 3476   ~Pcpw 4010   dom cdm 4999   ` cfv 5588   GrpOpcgr 24892   SubGrpOpcsubgo 25007
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 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
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 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fv 5596  df-subgo 25008
This theorem is referenced by:  subgores  25010  subgoid  25013  subgoinv  25014  issubgoi  25016  subgoablo  25017  ghsubgolem  25076  hhssabloi  25882  ghomfo  28534  ghomgsg  28536
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