Theorem subgres 8113

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		fun2ssres 3559	. . 3 |- ((Fun G /\ H (_ G /\ (W X. W) (_ dom H) -> (G |` (W X. W)) = (H |` (W X. W)))
2		issubg 8112	. . . . . 6 |- (H e. (SubGrp` G) <-> (G e. Grp /\ H e. Grp /\ H (_ G))
3	2	biimp 151	. . . . 5 |- (H e. (SubGrp` G) -> (G e. Grp /\ H e. Grp /\ H (_ G))
4	3	3simp1d 796	. . . 4 |- (H e. (SubGrp` G) -> G e. Grp)
5		eqid 1478	. . . . . 6 |- ran G = ran G
6	5	grpfo 8040	. . . . 5 |- (G e. Grp -> G:(ran G X. ran G)-onto->ran G)
7		fofun 3679	. . . . 5 |- (G:(ran G X. ran G)-onto->ran G -> Fun G)
8	6, 7	syl 10	. . . 4 |- (G e. Grp -> Fun G)
9	4, 8	syl 10	. . 3 |- (H e. (SubGrp` G) -> Fun G)
10	3	3simp3d 798	. . 3 |- (H e. (SubGrp` G) -> H (_ G)
11	3	3simp2d 797	. . . . 5 |- (H e. (SubGrp` G) -> H e. Grp)
12		subgres.1	. . . . . 6 |- W = ran H
13	12	grpfo 8040	. . . . 5 |- (H e. Grp -> H:(W X. W)-onto->W)
14		fof 3678	. . . . 5 |- (H:(W X. W)-onto->W -> H:(W X. W)-->W)
15	11, 13, 14	3syl 20	. . . 4 |- (H e. (SubGrp` G) -> H:(W X. W)-->W)
16		fdm 3637	. . . 4 |- (H:(W X. W)-->W -> dom H = (W X. W))
17		eqimss2 2113	. . . 4 |- (dom H = (W X. W) -> (W X. W) (_ dom H)
18	15, 16, 17	3syl 20	. . 3 |- (H e. (SubGrp` G) -> (W X. W) (_ dom H)
19	1, 9, 10, 18	syl3anc 860	. 2 |- (H e. (SubGrp` G) -> (G |` (W X. W)) = (H |` (W X. W)))
20		ffn 3633	. . . 4 |- (H:(W X. W)-->W -> H Fn (W X. W))
21		fnresdm 3602	. . . 4 |- (H Fn (W X. W) -> (H |` (W X. W)) = H)
22	14, 20, 21	3syl 20	. . 3 |- (H:(W X. W)-onto->W -> (H |` (W X. W)) = H)
23	11, 13, 22	3syl 20	. 2 |- (H e. (SubGrp` G) -> (H |` (W X. W)) = H)
24	19, 23	eqtr2d 1511	1 |- (H e. (SubGrp` G) -> H = (G |` (W X. W)))

Syntax hints:   -> wi 3   /\ w3a 777   = wceq 958   e. wcel 960   (_ wss 2050   X. cxp 3174  dom cdm 3176  ran crn 3177   |` cres 3178  Fun wfun 3182   Fn wfn 3183  -->wf 3184  -onto->wfo 3186  ` cfv 3188  Grpcgr 8030  SubGrpcsubg 8110

This theorem is referenced by:  subgopr 8114  subgrnss 8115

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-rab 1655  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-fo 3202  df-fv 3204  df-opr 3971  df-grp 8034  df-subg 8111

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