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Theorem issubgo 21844
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 3522 . . . . . . 7  |-  ( GrpOp  i^i 
~P G )  C_  ~P G
2 pwexg 4343 . . . . . . 7  |-  ( G  e.  GrpOp  ->  ~P G  e.  _V )
3 ssexg 4309 . . . . . . 7  |-  ( ( ( GrpOp  i^i  ~P G
)  C_  ~P G  /\  ~P G  e.  _V )  ->  ( GrpOp  i^i  ~P G )  e.  _V )
41, 2, 3sylancr 645 . . . . . 6  |-  ( G  e.  GrpOp  ->  ( GrpOp  i^i 
~P G )  e. 
_V )
5 pweq 3762 . . . . . . . 8  |-  ( g  =  G  ->  ~P g  =  ~P G
)
65ineq2d 3502 . . . . . . 7  |-  ( g  =  G  ->  ( GrpOp  i^i  ~P g )  =  ( GrpOp  i^i  ~P G ) )
7 df-subgo 21843 . . . . . . 7  |-  SubGrpOp  =  ( g  e.  GrpOp  |->  ( GrpOp  i^i 
~P g ) )
86, 7fvmptg 5763 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  ( GrpOp  i^i  ~P G )  e.  _V )  -> 
( SubGrpOp `  G )  =  ( GrpOp  i^i  ~P G ) )
94, 8mpdan 650 . . . . 5  |-  ( G  e.  GrpOp  ->  ( SubGrpOp `  G )  =  (
GrpOp  i^i  ~P G ) )
109eleq2d 2471 . . . 4  |-  ( G  e.  GrpOp  ->  ( H  e.  ( SubGrpOp `  G )  <->  H  e.  ( GrpOp  i^i  ~P G ) ) )
11 elin 3490 . . . . 5  |-  ( H  e.  ( GrpOp  i^i  ~P G )  <->  ( H  e.  GrpOp  /\  H  e.  ~P G ) )
12 elpwg 3766 . . . . . 6  |-  ( H  e.  GrpOp  ->  ( H  e.  ~P G  <->  H  C_  G
) )
1312pm5.32i 619 . . . . 5  |-  ( ( H  e.  GrpOp  /\  H  e.  ~P G )  <->  ( H  e.  GrpOp  /\  H  C_  G
) )
1411, 13bitri 241 . . . 4  |-  ( H  e.  ( GrpOp  i^i  ~P G )  <->  ( H  e.  GrpOp  /\  H  C_  G
) )
1510, 14syl6bb 253 . . 3  |-  ( G  e.  GrpOp  ->  ( H  e.  ( SubGrpOp `  G )  <->  ( H  e.  GrpOp  /\  H  C_  G ) ) )
1615pm5.32i 619 . 2  |-  ( ( G  e.  GrpOp  /\  H  e.  ( SubGrpOp `  G )
)  <->  ( G  e. 
GrpOp  /\  ( H  e. 
GrpOp  /\  H  C_  G
) ) )
177dmmptss 5325 . . . 4  |-  dom  SubGrpOp  C_  GrpOp
18 elfvdm 5716 . . . 4  |-  ( H  e.  ( SubGrpOp `  G
)  ->  G  e.  dom 
SubGrpOp )
1917, 18sseldi 3306 . . 3  |-  ( H  e.  ( SubGrpOp `  G
)  ->  G  e.  GrpOp
)
2019pm4.71ri 615 . 2  |-  ( H  e.  ( SubGrpOp `  G
)  <->  ( G  e. 
GrpOp  /\  H  e.  (
SubGrpOp `  G ) ) )
21 3anass 940 . 2  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  H  C_  G
)  <->  ( G  e. 
GrpOp  /\  ( H  e. 
GrpOp  /\  H  C_  G
) ) )
2216, 20, 213bitr4i 269 1  |-  ( H  e.  ( SubGrpOp `  G
)  <->  ( G  e. 
GrpOp  /\  H  e.  GrpOp  /\  H  C_  G )
)
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   _Vcvv 2916    i^i cin 3279    C_ wss 3280   ~Pcpw 3759   dom cdm 4837   ` cfv 5413   GrpOpcgr 21727   SubGrpOpcsubgo 21842
This theorem is referenced by:  subgores  21845  subgoid  21848  subgoinv  21849  issubgoi  21851  subgoablo  21852  ghsubgolem  21911  hhssabloi  22715  ghomfo  25055  ghomgsg  25057
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fv 5421  df-subgo 21843
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