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Theorem subgores 26025
Description: A subgroup operation is the restriction of its parent group operation to its underlying set. (Contributed by Paul Chapman, 3-Mar-2008.) (New usage is discouraged.)
Hypothesis
Ref Expression
subgores.1  |-  W  =  ran  H
Assertion
Ref Expression
subgores  |-  ( H  e.  ( SubGrpOp `  G
)  ->  H  =  ( G  |`  ( W  X.  W ) ) )

Proof of Theorem subgores
StepHypRef Expression
1 issubgo 26024 . . . . 5  |-  ( H  e.  ( SubGrpOp `  G
)  <->  ( G  e. 
GrpOp  /\  H  e.  GrpOp  /\  H  C_  G )
)
21simp1bi 1022 . . . 4  |-  ( H  e.  ( SubGrpOp `  G
)  ->  G  e.  GrpOp
)
3 eqid 2450 . . . . 5  |-  ran  G  =  ran  G
43grpofo 25920 . . . 4  |-  ( G  e.  GrpOp  ->  G :
( ran  G  X.  ran  G ) -onto-> ran  G
)
5 fofun 5792 . . . 4  |-  ( G : ( ran  G  X.  ran  G ) -onto-> ran 
G  ->  Fun  G )
62, 4, 53syl 18 . . 3  |-  ( H  e.  ( SubGrpOp `  G
)  ->  Fun  G )
71simp3bi 1024 . . 3  |-  ( H  e.  ( SubGrpOp `  G
)  ->  H  C_  G
)
81simp2bi 1023 . . . . 5  |-  ( H  e.  ( SubGrpOp `  G
)  ->  H  e.  GrpOp
)
9 subgores.1 . . . . . 6  |-  W  =  ran  H
109grpofo 25920 . . . . 5  |-  ( H  e.  GrpOp  ->  H :
( W  X.  W
) -onto-> W )
11 fof 5791 . . . . 5  |-  ( H : ( W  X.  W ) -onto-> W  ->  H : ( W  X.  W ) --> W )
128, 10, 113syl 18 . . . 4  |-  ( H  e.  ( SubGrpOp `  G
)  ->  H :
( W  X.  W
) --> W )
13 fdm 5731 . . . 4  |-  ( H : ( W  X.  W ) --> W  ->  dom  H  =  ( W  X.  W ) )
14 eqimss2 3484 . . . 4  |-  ( dom 
H  =  ( W  X.  W )  -> 
( W  X.  W
)  C_  dom  H )
1512, 13, 143syl 18 . . 3  |-  ( H  e.  ( SubGrpOp `  G
)  ->  ( W  X.  W )  C_  dom  H )
16 fun2ssres 5622 . . 3  |-  ( ( Fun  G  /\  H  C_  G  /\  ( W  X.  W )  C_  dom  H )  ->  ( G  |`  ( W  X.  W ) )  =  ( H  |`  ( W  X.  W ) ) )
176, 7, 15, 16syl3anc 1267 . 2  |-  ( H  e.  ( SubGrpOp `  G
)  ->  ( G  |`  ( W  X.  W
) )  =  ( H  |`  ( W  X.  W ) ) )
18 fofn 5793 . . 3  |-  ( H : ( W  X.  W ) -onto-> W  ->  H  Fn  ( W  X.  W ) )
19 fnresdm 5683 . . 3  |-  ( H  Fn  ( W  X.  W )  ->  ( H  |`  ( W  X.  W ) )  =  H )
208, 10, 18, 194syl 19 . 2  |-  ( H  e.  ( SubGrpOp `  G
)  ->  ( H  |`  ( W  X.  W
) )  =  H )
2117, 20eqtr2d 2485 1  |-  ( H  e.  ( SubGrpOp `  G
)  ->  H  =  ( G  |`  ( W  X.  W ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1443    e. wcel 1886    C_ wss 3403    X. cxp 4831   dom cdm 4833   ran crn 4834    |` cres 4835   Fun wfun 5575    Fn wfn 5576   -->wf 5577   -onto->wfo 5579   ` cfv 5581   GrpOpcgr 25907   SubGrpOpcsubgo 26022
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-op 3974  df-uni 4198  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-id 4748  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-fo 5587  df-fv 5589  df-ov 6291  df-grpo 25912  df-subgo 26023
This theorem is referenced by:  subgoov  26026  subgornss  26027
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