Theorem subgres 9426

Description: A subgroup operation is the restriction of its parent group operation to its underlying set. (Contributed by Paul Chapman, 3-Mar-2008.)

Hypothesis

Ref	Expression
subgres.1	|- W = ran H

Assertion

Ref	Expression
subgres	|- (H e. (SubGrp` G) -> H = (G |` (W X. W)))

Proof of Theorem subgres

Step	Hyp	Ref	Expression
1		issubg 9425	. . . . 5 |- (H e. (SubGrp` G) <-> (G e. Grp /\ H e. Grp /\ H C_ G))
2	1	simp1bi 891	. . . 4 |- (H e. (SubGrp` G) -> G e. Grp)
3		eqid 1884	. . . . . 6 |- ran G = ran G
4	3	grpfo 9323	. . . . 5 |- (G e. Grp -> G:(ran G X. ran G)-onto->ran G)
5		fofun 4618	. . . . 5 |- (G:(ran G X. ran G)-onto->ran G -> Fun G)
6	4, 5	syl 12	. . . 4 |- (G e. Grp -> Fun G)
7	2, 6	syl 12	. . 3 |- (H e. (SubGrp` G) -> Fun G)
8	1	simp3bi 893	. . 3 |- (H e. (SubGrp` G) -> H C_ G)
9	1	simp2bi 892	. . . . 5 |- (H e. (SubGrp` G) -> H e. Grp)
10		subgres.1	. . . . . 6 |- W = ran H
11	10	grpfo 9323	. . . . 5 |- (H e. Grp -> H:(W X. W)-onto->W)
12		fof 4617	. . . . 5 |- (H:(W X. W)-onto->W -> H:(W X. W)-->W)
13	9, 11, 12	3syl 24	. . . 4 |- (H e. (SubGrp` G) -> H:(W X. W)-->W)
14		fdm 4567	. . . 4 |- (H:(W X. W)-->W -> dom H = (W X. W))
15		eqimss2 2667	. . . 4 |- (dom H = (W X. W) -> (W X. W) C_ dom H)
16	13, 14, 15	3syl 24	. . 3 |- (H e. (SubGrp` G) -> (W X. W) C_ dom H)
17		fun2ssres 4461	. . 3 |- ((Fun G /\ H C_ G /\ (W X. W) C_ dom H) -> (G |` (W X. W)) = (H |` (W X. W)))
18	7, 8, 16, 17	syl111anc 1100	. 2 |- (H e. (SubGrp` G) -> (G |` (W X. W)) = (H |` (W X. W)))
19		fofn 4619	. . . 4 |- (H:(W X. W)-onto->W -> H Fn (W X. W))
20		fnresdm 4522	. . . 4 |- (H Fn (W X. W) -> (H |` (W X. W)) = H)
21	19, 20	syl 12	. . 3 |- (H:(W X. W)-onto->W -> (H |` (W X. W)) = H)
22	9, 11, 21	3syl 24	. 2 |- (H e. (SubGrp` G) -> (H |` (W X. W)) = H)
23	18, 22	eqtr2d 1926	1 |- (H e. (SubGrp` G) -> H = (G |` (W X. W)))

Syntax hints:   -> wi 3   = wceq 1298   e. wcel 1300   C_ wss 2593   X. cxp 3984  dom cdm 3986  ran crn 3987   |` cres 3988  Fun wfun 3992   Fn wfn 3993  -->wf 3994  -onto->wfo 3996  ` cfv 3998  Grpcgr 9311  SubGrpcsubg 9423

This theorem is referenced by:  subgopr 9427  subgrnss 9428

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-fo 4012  df-fv 4014  df-opr 4886  df-grp 9316  df-subg 9424

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