Theorem ghomgrp 10385

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 1478	. . 3 |- {<.<.x, x>., x>.} = {<.<.x, x>., x>.}
3		eqid 1478	. . 3 |- (I |` {x}) = (I |` {x})
4	1, 2, 3	ghomgrplem 10384	. 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 3341	. . . . 5 |- ((Y X. Y) = (ran F X. ran F) <-> Y = ran F)
8		reseq2 3375	. . . . 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 1498	. 2 |- S = (H |` (ran F X. ran F))
12	4, 11	syl5eqel 1555	1 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> S e. (SubGrp` H))

Syntax hints:   -> wi 3   /\ w3a 777   = wceq 958   e. wcel 960  {csn 2413  <.cop 2415  Icid 2837   X. cxp 3174  ran crn 3177   |` cres 3178  ` cfv 3188  (class class class)co 3969  Grpcgr 8030  SubGrpcsubg 8110   GrpHom cghom 10373

This theorem is referenced by:  ghomfo 10386  ghomgsg 10390

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-rep 2698  ax-sep 2708  ax-nul 2715  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-reu 1654  df-rab 1655  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-if 2366  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-f1 3201  df-fo 3202  df-f1o 3203  df-fv 3204  df-opr 3971  df-oprab 3972  df-grp 8034  df-gid 8035  df-ginv 8036  df-subg 8111  df-ghom 10375

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