Theorem ghomfo 13634

Description: A group homomorphism maps onto its image. (Contributed by Paul Chapman, 3-Mar-2008.)

Hypotheses

Ref	Expression
ghomfo.1	|- X = ran G
ghomfo.2	|- Y = ran F
ghomfo.3	|- S = (H |` (Y X. Y))
ghomfo.4	|- Z = ran S

Assertion

Ref	Expression
ghomfo	|- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> F:X-onto->Z)

Proof of Theorem ghomfo

Step	Hyp	Ref	Expression
1		df-fo 4012	. 2 |- (F:X-onto->Z <-> (F Fn X /\ ran F = Z))
2		ghomfo.1	. . . . . 6 |- X = ran G
3		eqid 1884	. . . . . 6 |- ran H = ran H
4	2, 3	elghom 10195	. . . . 5 |- ((G e. Grp /\ H e. Grp) -> (F e. (G GrpHom H) <-> (F:X-->ran H /\ A.x e. X A.y e. X ((F` x)H(F` y)) = (F` (xGy)))))
5	4	biimp3a 1194	. . . 4 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> (F:X-->ran H /\ A.x e. X A.y e. X ((F` x)H(F` y)) = (F` (xGy))))
6	5	simplld 348	. . 3 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> F:X-->ran H)
7		ffn 4562	. . 3 |- (F:X-->ran H -> F Fn X)
8	6, 7	syl 12	. 2 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> F Fn X)
9		ghomfo.2	. . . . . . . . . 10 |- Y = ran F
10		ghomfo.3	. . . . . . . . . 10 |- S = (H |` (Y X. Y))
11	9, 10	ghomgrp 13633	. . . . . . . . 9 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> S e. (SubGrp` H))
12		issubg 9425	. . . . . . . . 9 |- (S e. (SubGrp` H) <-> (H e. Grp /\ S e. Grp /\ S C_ H))
13	11, 12	sylib 215	. . . . . . . 8 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> (H e. Grp /\ S e. Grp /\ S C_ H))
14	13	simp2d 889	. . . . . . 7 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> S e. Grp)
15		ghomfo.4	. . . . . . . . 9 |- Z = ran S
16	15	grpfo 9323	. . . . . . . 8 |- (S e. Grp -> S:(Z X. Z)-onto->Z)
17		fof 4617	. . . . . . . 8 |- (S:(Z X. Z)-onto->Z -> S:(Z X. Z)-->Z)
18		fdm 4567	. . . . . . . 8 |- (S:(Z X. Z)-->Z -> dom S = (Z X. Z))
19	16, 17, 18	3syl 24	. . . . . . 7 |- (S e. Grp -> dom S = (Z X. Z))
20	14, 19	syl 12	. . . . . 6 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> dom S = (Z X. Z))
21	10	dmeqi 4158	. . . . . 6 |- dom S = dom ( H |` (Y X. Y))
22	20, 21	syl5reqr 1943	. . . . 5 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> (Z X. Z) = dom ( H |` (Y X. Y)))
23		frn 4569	. . . . . . . . 9 |- (F:X-->ran H -> ran F C_ ran H)
24	6, 23	syl 12	. . . . . . . 8 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> ran F C_ ran H)
25	24, 9	syl5ss 2661	. . . . . . 7 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> Y C_ ran H)
26		xpss12 4089	. . . . . . 7 |- ((Y C_ ran H /\ Y C_ ran H) -> (Y X. Y) C_ (ran H X. ran H))
27	25, 25, 26	syl11anc 524	. . . . . 6 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> (Y X. Y) C_ (ran H X. ran H))
28	3	grpfo 9323	. . . . . . . . . 10 |- (H e. Grp -> H:(ran H X. ran H)-onto->ran H)
29		fof 4617	. . . . . . . . . 10 |- (H:(ran H X. ran H)-onto->ran H -> H:(ran H X. ran H)-->ran H)
30		fdm 4567	. . . . . . . . . 10 |- (H:(ran H X. ran H)-->ran H -> dom H = (ran H X. ran H))
31	28, 29, 30	3syl 24	. . . . . . . . 9 |- (H e. Grp -> dom H = (ran H X. ran H))
32	31	sseq2d 2645	. . . . . . . 8 |- (H e. Grp -> ((Y X. Y) C_ dom H <-> (Y X. Y) C_ (ran H X. ran H)))
33		ssdmres 4235	. . . . . . . 8 |- ((Y X. Y) C_ dom H <-> dom ( H |` (Y X. Y)) = (Y X. Y))
34	32, 33	syl5rbbr 594	. . . . . . 7 |- (H e. Grp -> ((Y X. Y) C_ (ran H X. ran H) <-> dom ( H |` (Y X. Y)) = (Y X. Y)))
35	34	3ad2ant2 898	. . . . . 6 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> ((Y X. Y) C_ (ran H X. ran H) <-> dom ( H |` (Y X. Y)) = (Y X. Y)))
36	27, 35	mpbid 212	. . . . 5 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> dom ( H |` (Y X. Y)) = (Y X. Y))
37	22, 36	eqtrd 1925	. . . 4 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> (Z X. Z) = (Y X. Y))
38		xpid11 4181	. . . 4 |- ((Z X. Z) = (Y X. Y) <-> Z = Y)
39	37, 38	sylib 215	. . 3 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> Z = Y)
40	39, 9	syl6req 1945	. 2 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> ran F = Z)
41	1, 8, 40	sylanbrc 527	1 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> F:X-onto->Z)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105   C_ wss 2593   X. cxp 3984  dom cdm 3986  ran crn 3987   |` cres 3988   Fn wfn 3993  -->wf 3994  -onto->wfo 3996  ` cfv 3998  (class class class)co 4884  Grpcgr 9311  SubGrpcsubg 9423   GrpHom cghom 10189

This theorem is referenced by:  ghomcl 13635  ghomgsg 13636  ghomf1olem 13637  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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