Theorem ghomgsg 10390

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 1478	. . . 4 |- ran G = ran G
2		ghomgsg.1	. . . 4 |- Y = ran F
3		ghomgsg.2	. . . 4 |- S = (H |` (Y X. Y))
4		eqid 1478	. . . 4 |- ran S = ran S
5	1, 2, 3, 4	ghomfo 10386	. . 3 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> F:ran G-onto->ran S)
6		fof 3678	. . 3 |- (F:ran G-onto->ran S -> F:ran G-->ran S)
7	5, 6	syl 10	. 2 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> F:ran G-->ran S)
8		eqid 1478	. . . . . 6 |- ran H = ran H
9	1, 8	elghom 10379	. . . . 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 921	. . . 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	pm3.27d 325	. . 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 10385	. . . . . . . 8 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> S e. (SubGrp` H))
13	4	subgopr 8114	. . . . . . . 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 10	. . . . . . 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 10387	. . . . . . 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 10387	. . . . . . 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 461	. . . . . 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 350	. . . . 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 1486	. . . 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 1681	. . 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 196	. 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 10379	. . . . 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 154	. . . 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 801	. . 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 8112	. . . . 5 |- (S e. (SubGrp` H) <-> (H e. Grp /\ S e. Grp /\ S (_ H))
26	12, 25	sylib 198	. . . 4 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> (H e. Grp /\ S e. Grp /\ S (_ H))
27	26	3simp2d 797	. . 3 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> S e. Grp)
28	24, 27	syld3an2 874	. 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 705	1 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> F e. (G GrpHom S))

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 777   = wceq 958   e. wcel 960  A.wral 1648   (_ wss 2050   X. cxp 3174  ran crn 3177   |` cres 3178  -->wf 3184  -onto->wfo 3186  ` cfv 3188  (class class class)co 3969  Grpcgr 8030  SubGrpcsubg 8110   GrpHom cghom 10373

This theorem is referenced by:  cayleylem3 10406

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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