Theorem ghomgrp 10381

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 10380	. 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 3332	. . . . 5 |- ((Y X. Y) = (ran F X. ran F) <-> Y = ran F)
8		reseq2 3366	. . . . 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 1494	. 2 |- S = (H |` (ran F X. ran F))
12	4, 11	syl5eqel 1551	1 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> S e. (SubGrp` H))

Syntax hints:   -> wi 3   /\ w3a 774   = wceq 955   e. wcel 957  {csn 2407  <.cop 2409  Icid 2828   X. cxp 3165  ran crn 3168   |` cres 3169  ` cfv 3179  (class class class)co 3960  Grpcgr 8016  SubGrpcsubg 8099   GrpHom cghom 10369

This theorem is referenced by:  ghomfo 10382  ghomgsg 10386

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2690  ax-sep 2700  ax-nul 2707  ax-pow 2739  ax-pr 2776  ax-un 2863

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 980  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1586  df-ral 1648  df-rex 1649  df-reu 1650  df-rab 1651  df-v 1810  df-sbc 1940  df-csb 2000  df-dif 2047  df-un 2048  df-in 2049  df-ss 2051  df-nul 2279  df-if 2360  df-pw 2400  df-sn 2410  df-pr 2411  df-op 2414  df-uni 2501  df-br 2617  df-opab 2664  df-id 2832  df-xp 3181  df-rel 3182  df-cnv 3183  df-co 3184  df-dm 3185  df-rn 3186  df-res 3187  df-ima 3188  df-fun 3189  df-fn 3190  df-f 3191  df-f1 3192  df-fo 3193  df-f1o 3194  df-fv 3195  df-opr 3962  df-oprab 3963  df-grp 8020  df-gid 8021  df-ginv 8022  df-subg 8100  df-ghom 10371

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