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

Step	Hyp	Ref	Expression
1		issubgo 25902	. . . . 5 |- ( H e. ( SubGrpOp ` G ) <-> ( G e. GrpOp /\ H e. GrpOp /\ H C_ G ) )
2	1	simp1bi 1020	. . . 4 |- ( H e. ( SubGrpOp ` G ) -> G e. GrpOp )
3		eqid 2420	. . . . 5 |- ran G = ran G
4	3	grpofo 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 )
6	2, 4, 5	3syl 18	. . 3 |- ( H e. ( SubGrpOp ` G ) -> Fun G )
7	1	simp3bi 1022	. . 3 |- ( H e. ( SubGrpOp ` G ) -> H C_ G )
8	1	simp2bi 1021	. . . . 5 |- ( H e. ( SubGrpOp ` G ) -> H e. GrpOp )
9		subgores.1	. . . . . 6 |- W = ran H
10	9	grpofo 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 )
12	8, 10, 11	3syl 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 )
15	12, 13, 14	3syl 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 ) ) )
17	6, 7, 15, 16	syl3anc 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 )
20	8, 10, 18, 19	4syl 19	. 2 |- ( H e. ( SubGrpOp ` G ) -> ( H |` ( W X. W ) ) = H )
21	17, 20	eqtr2d 2462	1 |- ( H e. ( SubGrpOp ` G ) -> H = ( G |` ( W X. W ) ) )

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