Theorem ghomgrp 10390

Description: The image of a group homomorphism from G to H is a subgroup of H. (Contributed by Paul Chapman, 25-Feb-2008.)

Hypotheses

Ref	Expression
ghomgrp.1	|- Y = ran F
ghomgrp.2	|- S = (H |` (Y X. Y))

Assertion

Ref	Expression
ghomgrp	|- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> S e. (SubGrp` H))

Proof of Theorem ghomgrp

Step	Hyp	Ref	Expression
1		id 59	. . 3 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> (G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)))
2		eqid 1475	. . 3 |- {<.<.x, x>., x>.} = {<.<.x, x>., x>.}
3		eqid 1475	. . 3 |- (I |` {x}) = (I |` {x})
4	1, 2, 3	ghomgrplem 10389	. 2 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> (H |` (ran F X. ran F)) e. (SubGrp` H))
5		ghomgrp.2	. . 3 |- S = (H |` (Y X. Y))
6		ghomgrp.1	. . . 4 |- Y = ran F
7		xpid11 3335	. . . . 5 |- ((Y X. Y) = (ran F X. ran F) <-> Y = ran F)
8		reseq2 3369	. . . . 5 |- ((Y X. Y) = (ran F X. ran F) -> (H |` (Y X. Y)) = (H |` (ran F X. ran F)))
9	7, 8	sylbir 201	. . . 4 |- (Y = ran F -> (H |` (Y X. Y)) = (H |` (ran F X. ran F)))
10	6, 9	ax-mp 7	. . 3 |- (H |` (Y X. Y)) = (H |` (ran F X. ran F))
11	5, 10	eqtr 1495	. 2 |- S = (H |` (ran F X. ran F))
12	4, 11	syl5eqel 1552	1 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> S e. (SubGrp` H))

Syntax hints:   -> wi 3   /\ w3a 775   = wceq 956   e. wcel 958  {csn 2409  <.cop 2411  Icid 2831   X. cxp 3168  ran crn 3171   |` cres 3172  ` cfv 3182  (class class class)co 3963  Grpcgr 8033  SubGrpcsubg 8114   GrpHom cghom 10378

This theorem is referenced by:  ghomfo 10391  ghomgsg 10395

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-rep 2693  ax-sep 2703  ax-nul 2710  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-reu 1651  df-rab 1652  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-if 2362  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-f1 3195  df-fo 3196  df-f1o 3197  df-fv 3198  df-opr 3965  df-oprab 3966  df-grp 8037  df-gid 8038  df-ginv 8039  df-subg 8115  df-ghom 10380

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