Theorem ghomgsg 10395

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 1475	. . . 4 |- ran G = ran G
2		ghomgsg.1	. . . 4 |- Y = ran F
3		ghomgsg.2	. . . 4 |- S = (H |` (Y X. Y))
4		eqid 1475	. . . 4 |- ran S = ran S
5	1, 2, 3, 4	ghomfo 10391	. . 3 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> F:ran G-onto->ran S)
6		fof 3672	. . 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 1475	. . . . . 6 |- ran H = ran H
9	1, 8	elghom 10384	. . . . 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 919	. . . 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 10390	. . . . . . . 8 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> S e. (SubGrp` H))
13	4	subgopr 8118	. . . . . . . 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 10392	. . . . . . 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 10392	. . . . . . 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 459	. . . . . 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 1483	. . . 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 1678	. . 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 10384	. . . . 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 799	. . 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 8116	. . . . 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 795	. . 3 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> S e. Grp)
28	24, 27	syld3an2 872	. 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 703	1 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> F e. (G GrpHom S))

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 775   = wceq 956   e. wcel 958  A.wral 1645   (_ wss 2047   X. cxp 3168  ran crn 3171   |` cres 3172  -->wf 3178  -onto->wfo 3180  ` cfv 3182  (class class class)co 3963  Grpcgr 8033  SubGrpcsubg 8114   GrpHom cghom 10378

This theorem is referenced by:  cayleylem3 10411

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

Copyright terms: Public domain