Theorem ghomgsg 13636

Description: A group homomorphism from G to H is also a group homomorphism from G to its image in H. (Contributed by Paul Chapman, 3-Mar-2008.)

Hypotheses

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

Assertion

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

Proof of Theorem ghomgsg

Step	Hyp	Ref	Expression
1		eqid 1884	. . . 4 |- ran G = ran G
2		ghomgsg.1	. . . 4 |- Y = ran F
3		ghomgsg.2	. . . 4 |- S = (H |` (Y X. Y))
4		eqid 1884	. . . 4 |- ran S = ran S
5	1, 2, 3, 4	ghomfo 13634	. . 3 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> F:ran G-onto->ran S)
6		fof 4617	. . 3 |- (F:ran G-onto->ran S -> F:ran G-->ran S)
7	5, 6	syl 12	. 2 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> F:ran G-->ran S)
8		eqid 1884	. . . . . 6 |- ran H = ran H
9	1, 8	elghom 10195	. . . . 5 |- ((G e. Grp /\ H e. Grp) -> (F e. (G GrpHom H) <-> (F:ran G-->ran H /\ A.x e. ran GA.y e. ran G((F` x)H(F` y)) = (F` (xGy)))))
10	9	biimp3a 1194	. . . 4 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> (F:ran G-->ran H /\ A.x e. ran GA.y e. ran G((F` x)H(F` y)) = (F` (xGy))))
11	10	simprd 352	. . 3 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> A.x e. ran GA.y e. ran G((F` x)H(F` y)) = (F` (xGy)))
12	2, 3	ghomgrp 13633	. . . . . . . 8 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> S e. (SubGrp` H))
13	4	subgopr 9427	. . . . . . . 8 |- (S e. (SubGrp` H) -> (((F` x) e. ran S /\ (F` y) e. ran S) -> ((F` x)S(F` y)) = ((F` x)H(F` y))))
14	12, 13	syl 12	. . . . . . 7 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> (((F` x) e. ran S /\ (F` y) e. ran S) -> ((F` x)S(F` y)) = ((F` x)H(F` y))))
15	1, 2, 3, 4	ghomcl 13635	. . . . . . 7 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> (x e. ran G -> (F` x) e. ran S))
16	1, 2, 3, 4	ghomcl 13635	. . . . . . 7 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> (y e. ran G -> (F` y) e. ran S))
17	14, 15, 16	syl2and 508	. . . . . 6 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> ((x e. ran G /\ y e. ran G) -> ((F` x)S(F` y)) = ((F` x)H(F` y))))
18	17	imp 377	. . . . 5 |- (((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) /\ (x e. ran G /\ y e. ran G)) -> ((F` x)S(F` y)) = ((F` x)H(F` y)))
19	18	eqeq1d 1892	. . . 4 |- (((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) /\ (x e. ran G /\ y e. ran G)) -> (((F` x)S(F` y)) = (F` (xGy)) <-> ((F` x)H(F` y)) = (F` (xGy))))
20	19	2ralbidva 2138	. . 3 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> (A.x e. ran GA.y e. ran G((F` x)S(F` y)) = (F` (xGy)) <-> A.x e. ran GA.y e. ran G((F` x)H(F` y)) = (F` (xGy))))
21	11, 20	mpbird 213	. 2 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> A.x e. ran GA.y e. ran G((F` x)S(F` y)) = (F` (xGy)))
22	1, 4	elghom 10195	. . . . 5 |- ((G e. Grp /\ S e. Grp) -> (F e. (G GrpHom S) <-> (F:ran G-->ran S /\ A.x e. ran GA.y e. ran G((F` x)S(F` y)) = (F` (xGy)))))
23	22	biimprd 171	. . . 4 |- ((G e. Grp /\ S e. Grp) -> ((F:ran G-->ran S /\ A.x e. ran GA.y e. ran G((F` x)S(F` y)) = (F` (xGy))) -> F e. (G GrpHom S)))
24	23	3adant3 896	. . 3 |- ((G e. Grp /\ S e. Grp /\ F e. (G GrpHom H)) -> ((F:ran G-->ran S /\ A.x e. ran GA.y e. ran G((F` x)S(F` y)) = (F` (xGy))) -> F e. (G GrpHom S)))
25		issubg 9425	. . . . 5 |- (S e. (SubGrp` H) <-> (H e. Grp /\ S e. Grp /\ S C_ H))
26	12, 25	sylib 215	. . . 4 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> (H e. Grp /\ S e. Grp /\ S C_ H))
27	26	simp2d 889	. . 3 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> S e. Grp)
28	24, 27	syld3an2 1144	. 2 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> ((F:ran G-->ran S /\ A.x e. ran GA.y e. ran G((F` x)S(F` y)) = (F` (xGy))) -> F e. (G GrpHom S)))
29	7, 21, 28	mp2and 767	1 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> F e. (G GrpHom S))

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105   C_ wss 2593   X. cxp 3984  ran crn 3987   |` cres 3988  -->wf 3994  -onto->wfo 3996  ` cfv 3998  (class class class)co 4884  Grpcgr 9311  SubGrpcsubg 9423   GrpHom cghom 10189

This theorem is referenced by:  cayleylem3 13643

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-grp 9316  df-gid 9317  df-ginv 9318  df-subg 9424  df-ghom 10190

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