Theorem ghomfo 10386

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 3202	. . 3 |- (F:X-onto->Z <-> (F Fn X /\ ran F = Z))
2	1	biimpr 152	. 2 |- ((F Fn X /\ ran F = Z) -> F:X-onto->Z)
3		ghomfo.1	. . . . . 6 |- X = ran G
4		eqid 1478	. . . . . 6 |- ran H = ran H
5	3, 4	elghom 10379	. . . . 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)))))
6	5	biimp3a 921	. . . 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))))
7	6	pm3.26d 321	. . 3 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> F:X-->ran H)
8		ffn 3633	. . 3 |- (F:X-->ran H -> F Fn X)
9	7, 8	syl 10	. 2 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> F Fn X)
10		ghomfo.2	. . . . . . . . . 10 |- Y = ran F
11		ghomfo.3	. . . . . . . . . 10 |- S = (H |` (Y X. Y))
12	10, 11	ghomgrp 10385	. . . . . . . . 9 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> S e. (SubGrp` H))
13		issubg 8112	. . . . . . . . 9 |- (S e. (SubGrp` H) <-> (H e. Grp /\ S e. Grp /\ S (_ H))
14	12, 13	sylib 198	. . . . . . . 8 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> (H e. Grp /\ S e. Grp /\ S (_ H))
15	14	3simp2d 797	. . . . . . 7 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> S e. Grp)
16		ghomfo.4	. . . . . . . . 9 |- Z = ran S
17	16	grpfo 8040	. . . . . . . 8 |- (S e. Grp -> S:(Z X. Z)-onto->Z)
18		fof 3678	. . . . . . . 8 |- (S:(Z X. Z)-onto->Z -> S:(Z X. Z)-->Z)
19		fdm 3637	. . . . . . . 8 |- (S:(Z X. Z)-->Z -> dom S = (Z X. Z))
20	17, 18, 19	3syl 20	. . . . . . 7 |- (S e. Grp -> dom S = (Z X. Z))
21	15, 20	syl 10	. . . . . 6 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> dom S = (Z X. Z))
22	11	dmeqi 3318	. . . . . 6 |- dom S = dom ( H |` (Y X. Y))
23	21, 22	syl5reqr 1525	. . . . 5 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> (Z X. Z) = dom ( H |` (Y X. Y)))
24		ssxp 3262	. . . . . . 7 |- ((Y (_ ran H /\ Y (_ ran H) -> (Y X. Y) (_ (ran H X. ran H))
25		frn 3639	. . . . . . . . 9 |- (F:X-->ran H -> ran F (_ ran H)
26	7, 25	syl 10	. . . . . . . 8 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> ran F (_ ran H)
27	26, 10	syl5ss 2108	. . . . . . 7 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> Y (_ ran H)
28	24, 27, 27	sylanc 473	. . . . . 6 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> (Y X. Y) (_ (ran H X. ran H))
29	4	grpfo 8040	. . . . . . . . . 10 |- (H e. Grp -> H:(ran H X. ran H)-onto->ran H)
30		fof 3678	. . . . . . . . . 10 |- (H:(ran H X. ran H)-onto->ran H -> H:(ran H X. ran H)-->ran H)
31		fdm 3637	. . . . . . . . . 10 |- (H:(ran H X. ran H)-->ran H -> dom H = (ran H X. ran H))
32	29, 30, 31	3syl 20	. . . . . . . . 9 |- (H e. Grp -> dom H = (ran H X. ran H))
33	32	sseq2d 2092	. . . . . . . 8 |- (H e. Grp -> ((Y X. Y) (_ dom H <-> (Y X. Y) (_ (ran H X. ran H)))
34		ssdmres 3387	. . . . . . . 8 |- ((Y X. Y) (_ dom H <-> dom ( H |` (Y X. Y)) = (Y X. Y))
35	33, 34	syl5rbbr 537	. . . . . . 7 |- (H e. Grp -> ((Y X. Y) (_ (ran H X. ran H) <-> dom ( H |` (Y X. Y)) = (Y X. Y)))
36	35	3ad2ant2 803	. . . . . 6 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> ((Y X. Y) (_ (ran H X. ran H) <-> dom ( H |` (Y X. Y)) = (Y X. Y)))
37	28, 36	mpbid 195	. . . . 5 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> dom ( H |` (Y X. Y)) = (Y X. Y))
38	23, 37	eqtrd 1510	. . . 4 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> (Z X. Z) = (Y X. Y))
39		xpid11 3341	. . . 4 |- ((Z X. Z) = (Y X. Y) <-> Z = Y)
40	38, 39	sylib 198	. . 3 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> Z = Y)
41	40, 10	syl6req 1527	. 2 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> ran F = Z)
42	2, 9, 41	sylanc 473	1 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> F:X-onto->Z)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 777   = wceq 958   e. wcel 960  A.wral 1648   (_ wss 2050   X. cxp 3174  dom cdm 3176  ran crn 3177   |` cres 3178   Fn wfn 3183  -->wf 3184  -onto->wfo 3186  ` cfv 3188  (class class class)co 3969  Grpcgr 8030  SubGrpcsubg 8110   GrpHom cghom 10373

This theorem is referenced by:  ghomcl 10387  ghomgsg 10390  ghomf1olem 10391  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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