Theorem subgrnss 8119

Description: The underlying set of a subgroup is a subset of its parent group's underlying set. (Contributed by Paul Chapman, 3-Mar-2008.)

Hypotheses

Ref	Expression
subgrnss.1	|- X = ran G
subgrnss.2	|- W = ran H

Assertion

Ref	Expression
subgrnss	|- (H e. (SubGrp` G) -> W (_ X)

Proof of Theorem subgrnss

Step	Hyp	Ref	Expression
1		imassrn 3415	. . 3 |- (G"(W X. W)) (_ ran G
2		subgrnss.2	. . . . . . 7 |- W = ran H
3	2	subgres 8117	. . . . . 6 |- (H e. (SubGrp` G) -> H = (G |` (W X. W)))
4	3	rneqd 3341	. . . . 5 |- (H e. (SubGrp` G) -> ran H = ran ( G |` (W X. W)))
5		df-ima 3191	. . . . 5 |- (G"(W X. W)) = ran ( G |` (W X. W))
6	4, 5	syl6eqr 1525	. . . 4 |- (H e. (SubGrp` G) -> ran H = (G"(W X. W)))
7	6	sseq1d 2088	. . 3 |- (H e. (SubGrp` G) -> (ran H (_ ran G <-> (G"(W X. W)) (_ ran G))
8	1, 7	mpbiri 194	. 2 |- (H e. (SubGrp` G) -> ran H (_ ran G)
9		subgrnss.1	. 2 |- X = ran G
10	8, 2, 9	3sstr4g 2102	1 |- (H e. (SubGrp` G) -> W (_ X)

Syntax hints:   -> wi 3   = wceq 956   e. wcel 958   (_ wss 2047   X. cxp 3168  ran crn 3171   |` cres 3172  "cima 3173  ` cfv 3182  SubGrpcsubg 8114

This theorem is referenced by:  subgid 8120  subgabl 8123  ghsubgi 8138

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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