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

Step	Hyp	Ref	Expression
1		issubgo 25078	. . . . 5 |- ( H e. ( SubGrpOp ` G ) <-> ( G e. GrpOp /\ H e. GrpOp /\ H C_ G ) )
2	1	simp1bi 1011	. . . 4 |- ( H e. ( SubGrpOp ` G ) -> G e. GrpOp )
3		eqid 2467	. . . . 5 |- ran G = ran G
4	3	grpofo 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 )
6	2, 4, 5	3syl 20	. . 3 |- ( H e. ( SubGrpOp ` G ) -> Fun G )
7	1	simp3bi 1013	. . 3 |- ( H e. ( SubGrpOp ` G ) -> H C_ G )
8	1	simp2bi 1012	. . . . 5 |- ( H e. ( SubGrpOp ` G ) -> H e. GrpOp )
9		subgores.1	. . . . . 6 |- W = ran H
10	9	grpofo 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 )
12	8, 10, 11	3syl 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 )
15	12, 13, 14	3syl 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 ) ) )
17	6, 7, 15, 16	syl3anc 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 )
20	8, 10, 18, 19	4syl 21	. 2 |- ( H e. ( SubGrpOp ` G ) -> ( H |` ( W X. W ) ) = H )
21	17, 20	eqtr2d 2509	1 |- ( H e. ( SubGrpOp ` G ) -> H = ( G |` ( W X. W ) ) )

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