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Theorem subgores 21894
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 21893 . . . . 5  |-  ( H  e.  ( SubGrpOp `  G
)  <->  ( G  e. 
GrpOp  /\  H  e.  GrpOp  /\  H  C_  G )
)
21simp1bi 973 . . . 4  |-  ( H  e.  ( SubGrpOp `  G
)  ->  G  e.  GrpOp
)
3 eqid 2438 . . . . 5  |-  ran  G  =  ran  G
43grpofo 21789 . . . 4  |-  ( G  e.  GrpOp  ->  G :
( ran  G  X.  ran  G ) -onto-> ran  G
)
5 fofun 5656 . . . 4  |-  ( G : ( ran  G  X.  ran  G ) -onto-> ran 
G  ->  Fun  G )
62, 4, 53syl 19 . . 3  |-  ( H  e.  ( SubGrpOp `  G
)  ->  Fun  G )
71simp3bi 975 . . 3  |-  ( H  e.  ( SubGrpOp `  G
)  ->  H  C_  G
)
81simp2bi 974 . . . . 5  |-  ( H  e.  ( SubGrpOp `  G
)  ->  H  e.  GrpOp
)
9 subgores.1 . . . . . 6  |-  W  =  ran  H
109grpofo 21789 . . . . 5  |-  ( H  e.  GrpOp  ->  H :
( W  X.  W
) -onto-> W )
11 fof 5655 . . . . 5  |-  ( H : ( W  X.  W ) -onto-> W  ->  H : ( W  X.  W ) --> W )
128, 10, 113syl 19 . . . 4  |-  ( H  e.  ( SubGrpOp `  G
)  ->  H :
( W  X.  W
) --> W )
13 fdm 5597 . . . 4  |-  ( H : ( W  X.  W ) --> W  ->  dom  H  =  ( W  X.  W ) )
14 eqimss2 3403 . . . 4  |-  ( dom 
H  =  ( W  X.  W )  -> 
( W  X.  W
)  C_  dom  H )
1512, 13, 143syl 19 . . 3  |-  ( H  e.  ( SubGrpOp `  G
)  ->  ( W  X.  W )  C_  dom  H )
16 fun2ssres 5496 . . 3  |-  ( ( Fun  G  /\  H  C_  G  /\  ( W  X.  W )  C_  dom  H )  ->  ( G  |`  ( W  X.  W ) )  =  ( H  |`  ( W  X.  W ) ) )
176, 7, 15, 16syl3anc 1185 . 2  |-  ( H  e.  ( SubGrpOp `  G
)  ->  ( G  |`  ( W  X.  W
) )  =  ( H  |`  ( W  X.  W ) ) )
18 fofn 5657 . . 3  |-  ( H : ( W  X.  W ) -onto-> W  ->  H  Fn  ( W  X.  W ) )
19 fnresdm 5556 . . 3  |-  ( H  Fn  ( W  X.  W )  ->  ( H  |`  ( W  X.  W ) )  =  H )
208, 10, 18, 194syl 20 . 2  |-  ( H  e.  ( SubGrpOp `  G
)  ->  ( H  |`  ( W  X.  W
) )  =  H )
2117, 20eqtr2d 2471 1  |-  ( H  e.  ( SubGrpOp `  G
)  ->  H  =  ( G  |`  ( W  X.  W ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726    C_ wss 3322    X. cxp 4878   dom cdm 4880   ran crn 4881    |` cres 4882   Fun wfun 5450    Fn wfn 5451   -->wf 5452   -onto->wfo 5454   ` cfv 5456   GrpOpcgr 21776   SubGrpOpcsubgo 21891
This theorem is referenced by:  subgoov  21895  subgornss  21896
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-fo 5462  df-fv 5464  df-ov 6086  df-grpo 21781  df-subgo 21892
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