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Theorem subgores 25903
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 25902 . . . . 5  |-  ( H  e.  ( SubGrpOp `  G
)  <->  ( G  e. 
GrpOp  /\  H  e.  GrpOp  /\  H  C_  G )
)
21simp1bi 1020 . . . 4  |-  ( H  e.  ( SubGrpOp `  G
)  ->  G  e.  GrpOp
)
3 eqid 2420 . . . . 5  |-  ran  G  =  ran  G
43grpofo 25798 . . . 4  |-  ( G  e.  GrpOp  ->  G :
( ran  G  X.  ran  G ) -onto-> ran  G
)
5 fofun 5802 . . . 4  |-  ( G : ( ran  G  X.  ran  G ) -onto-> ran 
G  ->  Fun  G )
62, 4, 53syl 18 . . 3  |-  ( H  e.  ( SubGrpOp `  G
)  ->  Fun  G )
71simp3bi 1022 . . 3  |-  ( H  e.  ( SubGrpOp `  G
)  ->  H  C_  G
)
81simp2bi 1021 . . . . 5  |-  ( H  e.  ( SubGrpOp `  G
)  ->  H  e.  GrpOp
)
9 subgores.1 . . . . . 6  |-  W  =  ran  H
109grpofo 25798 . . . . 5  |-  ( H  e.  GrpOp  ->  H :
( W  X.  W
) -onto-> W )
11 fof 5801 . . . . 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 5741 . . . 4  |-  ( H : ( W  X.  W ) --> W  ->  dom  H  =  ( W  X.  W ) )
14 eqimss2 3514 . . . 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 5633 . . 3  |-  ( ( Fun  G  /\  H  C_  G  /\  ( W  X.  W )  C_  dom  H )  ->  ( G  |`  ( W  X.  W ) )  =  ( H  |`  ( W  X.  W ) ) )
176, 7, 15, 16syl3anc 1264 . 2  |-  ( H  e.  ( SubGrpOp `  G
)  ->  ( G  |`  ( W  X.  W
) )  =  ( H  |`  ( W  X.  W ) ) )
18 fofn 5803 . . 3  |-  ( H : ( W  X.  W ) -onto-> W  ->  H  Fn  ( W  X.  W ) )
19 fnresdm 5694 . . 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 2462 1  |-  ( H  e.  ( SubGrpOp `  G
)  ->  H  =  ( G  |`  ( W  X.  W ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437    e. wcel 1867    C_ wss 3433    X. cxp 4843   dom cdm 4845   ran crn 4846    |` cres 4847   Fun wfun 5586    Fn wfn 5587   -->wf 5588   -onto->wfo 5590   ` cfv 5592   GrpOpcgr 25785   SubGrpOpcsubgo 25900
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-fo 5598  df-fv 5600  df-ov 6299  df-grpo 25790  df-subgo 25901
This theorem is referenced by:  subgoov  25904  subgornss  25905
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