Theorem subgores 26025

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 26024	. . . . 5 |- ( H e. ( SubGrpOp ` G ) <-> ( G e. GrpOp /\ H e. GrpOp /\ H C_ G ) )
2	1	simp1bi 1022	. . . 4 |- ( H e. ( SubGrpOp ` G ) -> G e. GrpOp )
3		eqid 2450	. . . . 5 |- ran G = ran G
4	3	grpofo 25920	. . . 4 |- ( G e. GrpOp -> G : ( ran G X. ran G ) -onto-> ran G )
5		fofun 5792	. . . 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 1024	. . 3 |- ( H e. ( SubGrpOp ` G ) -> H C_ G )
8	1	simp2bi 1023	. . . . 5 |- ( H e. ( SubGrpOp ` G ) -> H e. GrpOp )
9		subgores.1	. . . . . 6 |- W = ran H
10	9	grpofo 25920	. . . . 5 |- ( H e. GrpOp -> H : ( W X. W ) -onto-> W )
11		fof 5791	. . . . 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 5731	. . . 4 |- ( H : ( W X. W ) --> W -> dom H = ( W X. W ) )
14		eqimss2 3484	. . . 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 5622	. . 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 1267	. 2 |- ( H e. ( SubGrpOp ` G ) -> ( G |` ( W X. W ) ) = ( H |` ( W X. W ) ) )
18		fofn 5793	. . 3 |- ( H : ( W X. W ) -onto-> W -> H Fn ( W X. W ) )
19		fnresdm 5683	. . 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 2485	1 |- ( H e. ( SubGrpOp ` G ) -> H = ( G |` ( W X. W ) ) )

Syntax hints:   -> wi 4   = wceq 1443   e. wcel 1886   C_ wss 3403   X. cxp 4831   dom cdm 4833   ran crn 4834   |` cres 4835   Fun wfun 5575   Fn wfn 5576   -->wf 5577   -onto->wfo 5579   ` cfv 5581   GrpOpcgr 25907   SubGrpOpcsubgo 26022

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-op 3974  df-uni 4198  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-id 4748  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-fo 5587  df-fv 5589  df-ov 6291  df-grpo 25912  df-subgo 26023

This theorem is referenced by:  subgoov  26026  subgornss  26027