Theorem ghomfo 10391

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 3196	. . 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 1475	. . . . . 6 |- ran H = ran H
5	3, 4	elghom 10384	. . . . 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 919	. . . 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 3627	. . 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 10390	. . . . . . . . 9 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> S e. (SubGrp` H))
13		issubg 8116	. . . . . . . . 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 795	. . . . . . 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 8043	. . . . . . . 8 |- (S e. Grp -> S:(Z X. Z)-onto->Z)
18		fof 3672	. . . . . . . 8 |- (S:(Z X. Z)-onto->Z -> S:(Z X. Z)-->Z)
19		fdm 3631	. . . . . . . 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 3312	. . . . . 6 |- dom S = dom ( H |` (Y X. Y))
23	21, 22	syl5reqr 1522	. . . . 5 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> (Z X. Z) = dom ( H |` (Y X. Y)))
24		ssxp 3256	. . . . . . 7 |- ((Y (_ ran H /\ Y (_ ran H) -> (Y X. Y) (_ (ran H X. ran H))
25		frn 3633	. . . . . . . . 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 2105	. . . . . . 7 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> Y (_ ran H)
28	24, 27, 27	sylanc 471	. . . . . 6 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> (Y X. Y) (_ (ran H X. ran H))
29	4	grpfo 8043	. . . . . . . . . 10 |- (H e. Grp -> H:(ran H X. ran H)-onto->ran H)
30		fof 3672	. . . . . . . . . 10 |- (H:(ran H X. ran H)-onto->ran H -> H:(ran H X. ran H)-->ran H)
31		fdm 3631	. . . . . . . . . 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 2089	. . . . . . . 8 |- (H e. Grp -> ((Y X. Y) (_ dom H <-> (Y X. Y) (_ (ran H X. ran H)))
34		ssdmres 3381	. . . . . . . 8 |- ((Y X. Y) (_ dom H <-> dom ( H |` (Y X. Y)) = (Y X. Y))
35	33, 34	syl5rbbr 535	. . . . . . 7 |- (H e. Grp -> ((Y X. Y) (_ (ran H X. ran H) <-> dom ( H |` (Y X. Y)) = (Y X. Y)))
36	35	3ad2ant2 801	. . . . . 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 1507	. . . 4 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> (Z X. Z) = (Y X. Y))
39		xpid11 3335	. . . 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 1524	. 2 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> ran F = Z)
42	2, 9, 41	sylanc 471	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 775   = wceq 956   e. wcel 958  A.wral 1645   (_ wss 2047   X. cxp 3168  dom cdm 3170  ran crn 3171   |` cres 3172   Fn wfn 3177  -->wf 3178  -onto->wfo 3180  ` cfv 3182  (class class class)co 3963  Grpcgr 8033  SubGrpcsubg 8114   GrpHom cghom 10378

This theorem is referenced by:  ghomcl 10392  ghomgsg 10395  ghomf1olem 10396  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

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