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Theorem issubgo 21402
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 3478 . . . . . . 7  |-  ( GrpOp  i^i 
~P G )  C_  ~P G
2 pwexg 4296 . . . . . . 7  |-  ( G  e.  GrpOp  ->  ~P G  e.  _V )
3 ssexg 4262 . . . . . . 7  |-  ( ( ( GrpOp  i^i  ~P G
)  C_  ~P G  /\  ~P G  e.  _V )  ->  ( GrpOp  i^i  ~P G )  e.  _V )
41, 2, 3sylancr 644 . . . . . 6  |-  ( G  e.  GrpOp  ->  ( GrpOp  i^i 
~P G )  e. 
_V )
5 pweq 3717 . . . . . . . 8  |-  ( g  =  G  ->  ~P g  =  ~P G
)
65ineq2d 3458 . . . . . . 7  |-  ( g  =  G  ->  ( GrpOp  i^i  ~P g )  =  ( GrpOp  i^i  ~P G ) )
7 df-subgo 21401 . . . . . . 7  |-  SubGrpOp  =  ( g  e.  GrpOp  |->  ( GrpOp  i^i 
~P g ) )
86, 7fvmptg 5707 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  ( GrpOp  i^i  ~P G )  e.  _V )  -> 
( SubGrpOp `  G )  =  ( GrpOp  i^i  ~P G ) )
94, 8mpdan 649 . . . . 5  |-  ( G  e.  GrpOp  ->  ( SubGrpOp `  G )  =  (
GrpOp  i^i  ~P G ) )
109eleq2d 2433 . . . 4  |-  ( G  e.  GrpOp  ->  ( H  e.  ( SubGrpOp `  G )  <->  H  e.  ( GrpOp  i^i  ~P G ) ) )
11 elin 3446 . . . . 5  |-  ( H  e.  ( GrpOp  i^i  ~P G )  <->  ( H  e.  GrpOp  /\  H  e.  ~P G ) )
12 elpwg 3721 . . . . . 6  |-  ( H  e.  GrpOp  ->  ( H  e.  ~P G  <->  H  C_  G
) )
1312pm5.32i 618 . . . . 5  |-  ( ( H  e.  GrpOp  /\  H  e.  ~P G )  <->  ( H  e.  GrpOp  /\  H  C_  G
) )
1411, 13bitri 240 . . . 4  |-  ( H  e.  ( GrpOp  i^i  ~P G )  <->  ( H  e.  GrpOp  /\  H  C_  G
) )
1510, 14syl6bb 252 . . 3  |-  ( G  e.  GrpOp  ->  ( H  e.  ( SubGrpOp `  G )  <->  ( H  e.  GrpOp  /\  H  C_  G ) ) )
1615pm5.32i 618 . 2  |-  ( ( G  e.  GrpOp  /\  H  e.  ( SubGrpOp `  G )
)  <->  ( G  e. 
GrpOp  /\  ( H  e. 
GrpOp  /\  H  C_  G
) ) )
177dmmptss 5272 . . . 4  |-  dom  SubGrpOp  C_  GrpOp
18 elfvdm 5661 . . . 4  |-  ( H  e.  ( SubGrpOp `  G
)  ->  G  e.  dom 
SubGrpOp )
1917, 18sseldi 3264 . . 3  |-  ( H  e.  ( SubGrpOp `  G
)  ->  G  e.  GrpOp
)
2019pm4.71ri 614 . 2  |-  ( H  e.  ( SubGrpOp `  G
)  <->  ( G  e. 
GrpOp  /\  H  e.  (
SubGrpOp `  G ) ) )
21 3anass 939 . 2  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  H  C_  G
)  <->  ( G  e. 
GrpOp  /\  ( H  e. 
GrpOp  /\  H  C_  G
) ) )
2216, 20, 213bitr4i 268 1  |-  ( H  e.  ( SubGrpOp `  G
)  <->  ( G  e. 
GrpOp  /\  H  e.  GrpOp  /\  H  C_  G )
)
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    /\ w3a 935    = wceq 1647    e. wcel 1715   _Vcvv 2873    i^i cin 3237    C_ wss 3238   ~Pcpw 3714   dom cdm 4792   ` cfv 5358   GrpOpcgr 21285   SubGrpOpcsubgo 21400
This theorem is referenced by:  subgores  21403  subgoid  21406  subgoinv  21407  issubgoi  21409  subgoablo  21410  ghsubgolem  21469  hhssabloi  22273  ghomfo  24670  ghomgsg  24672
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fv 5366  df-subgo 21401
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