Theorem subgres 8117

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 3553	. . 3 |- ((Fun G /\ H (_ G /\ (W X. W) (_ dom H) -> (G |` (W X. W)) = (H |` (W X. W)))
2		issubg 8116	. . . . . 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 794	. . . 4 |- (H e. (SubGrp` G) -> G e. Grp)
5		eqid 1475	. . . . . 6 |- ran G = ran G
6	5	grpfo 8043	. . . . 5 |- (G e. Grp -> G:(ran G X. ran G)-onto->ran G)
7		fofun 3673	. . . . 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 796	. . 3 |- (H e. (SubGrp` G) -> H (_ G)
11	3	3simp2d 795	. . . . 5 |- (H e. (SubGrp` G) -> H e. Grp)
12		subgres.1	. . . . . 6 |- W = ran H
13	12	grpfo 8043	. . . . 5 |- (H e. Grp -> H:(W X. W)-onto->W)
14		fof 3672	. . . . 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 3631	. . . 4 |- (H:(W X. W)-->W -> dom H = (W X. W))
17		eqimss2 2110	. . . 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 858	. 2 |- (H e. (SubGrp` G) -> (G |` (W X. W)) = (H |` (W X. W)))
20		ffn 3627	. . . 4 |- (H:(W X. W)-->W -> H Fn (W X. W))
21		fnresdm 3596	. . . 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 1508	1 |- (H e. (SubGrp` G) -> H = (G |` (W X. W)))

Syntax hints:   -> wi 3   /\ w3a 775   = wceq 956   e. wcel 958   (_ wss 2047   X. cxp 3168  dom cdm 3170  ran crn 3171   |` cres 3172  Fun wfun 3176   Fn wfn 3177  -->wf 3178  -onto->wfo 3180  ` cfv 3182  Grpcgr 8033  SubGrpcsubg 8114

This theorem is referenced by:  subgopr 8118  subgrnss 8119

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-rab 1652  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-fo 3196  df-fv 3198  df-opr 3965  df-grp 8037  df-subg 8115

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