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Theorem subgores 25079
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 25078 . . . . 5  |-  ( H  e.  ( SubGrpOp `  G
)  <->  ( G  e. 
GrpOp  /\  H  e.  GrpOp  /\  H  C_  G )
)
21simp1bi 1011 . . . 4  |-  ( H  e.  ( SubGrpOp `  G
)  ->  G  e.  GrpOp
)
3 eqid 2467 . . . . 5  |-  ran  G  =  ran  G
43grpofo 24974 . . . 4  |-  ( G  e.  GrpOp  ->  G :
( ran  G  X.  ran  G ) -onto-> ran  G
)
5 fofun 5796 . . . 4  |-  ( G : ( ran  G  X.  ran  G ) -onto-> ran 
G  ->  Fun  G )
62, 4, 53syl 20 . . 3  |-  ( H  e.  ( SubGrpOp `  G
)  ->  Fun  G )
71simp3bi 1013 . . 3  |-  ( H  e.  ( SubGrpOp `  G
)  ->  H  C_  G
)
81simp2bi 1012 . . . . 5  |-  ( H  e.  ( SubGrpOp `  G
)  ->  H  e.  GrpOp
)
9 subgores.1 . . . . . 6  |-  W  =  ran  H
109grpofo 24974 . . . . 5  |-  ( H  e.  GrpOp  ->  H :
( W  X.  W
) -onto-> W )
11 fof 5795 . . . . 5  |-  ( H : ( W  X.  W ) -onto-> W  ->  H : ( W  X.  W ) --> W )
128, 10, 113syl 20 . . . 4  |-  ( H  e.  ( SubGrpOp `  G
)  ->  H :
( W  X.  W
) --> W )
13 fdm 5735 . . . 4  |-  ( H : ( W  X.  W ) --> W  ->  dom  H  =  ( W  X.  W ) )
14 eqimss2 3557 . . . 4  |-  ( dom 
H  =  ( W  X.  W )  -> 
( W  X.  W
)  C_  dom  H )
1512, 13, 143syl 20 . . 3  |-  ( H  e.  ( SubGrpOp `  G
)  ->  ( W  X.  W )  C_  dom  H )
16 fun2ssres 5629 . . 3  |-  ( ( Fun  G  /\  H  C_  G  /\  ( W  X.  W )  C_  dom  H )  ->  ( G  |`  ( W  X.  W ) )  =  ( H  |`  ( W  X.  W ) ) )
176, 7, 15, 16syl3anc 1228 . 2  |-  ( H  e.  ( SubGrpOp `  G
)  ->  ( G  |`  ( W  X.  W
) )  =  ( H  |`  ( W  X.  W ) ) )
18 fofn 5797 . . 3  |-  ( H : ( W  X.  W ) -onto-> W  ->  H  Fn  ( W  X.  W ) )
19 fnresdm 5690 . . 3  |-  ( H  Fn  ( W  X.  W )  ->  ( H  |`  ( W  X.  W ) )  =  H )
208, 10, 18, 194syl 21 . 2  |-  ( H  e.  ( SubGrpOp `  G
)  ->  ( H  |`  ( W  X.  W
) )  =  H )
2117, 20eqtr2d 2509 1  |-  ( H  e.  ( SubGrpOp `  G
)  ->  H  =  ( G  |`  ( W  X.  W ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767    C_ wss 3476    X. cxp 4997   dom cdm 4999   ran crn 5000    |` cres 5001   Fun wfun 5582    Fn wfn 5583   -->wf 5584   -onto->wfo 5586   ` cfv 5588   GrpOpcgr 24961   SubGrpOpcsubgo 25076
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  ax-un 6577
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-csb 3436  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-iun 4327  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-fn 5591  df-f 5592  df-fo 5594  df-fv 5596  df-ov 6288  df-grpo 24966  df-subgo 25077
This theorem is referenced by:  subgoov  25080  subgornss  25081
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