Theorem grpkerinj 16042

Description: A group homomorphism is injective if and only if its kernel is zero.

Hypotheses

Ref	Expression
grpkerinj.1	|- X = ran G
grpkerinj.2	|- W = (Id` G)
grpkerinj.3	|- Y = ran H
grpkerinj.4	|- U = (Id` H)

Assertion

Ref	Expression
grpkerinj	|- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> (F:X-1-1->Y <-> (`'F"{U}) = {W}))

Proof of Theorem grpkerinj

Step	Hyp	Ref	Expression
1		grpkerinj.2	. . . . . . . . 9 |- W = (Id` G)
2		grpkerinj.4	. . . . . . . . 9 |- U = (Id` H)
3	1, 2	ghomid 10197	. . . . . . . 8 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> (F` W) = U)
4	3	sneqd 3056	. . . . . . 7 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> {(F` W)} = {U})
5		grpkerinj.1	. . . . . . . . . 10 |- X = ran G
6		grpkerinj.3	. . . . . . . . . 10 |- Y = ran H
7	5, 6	ghomf 16039	. . . . . . . . 9 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> F:X-->Y)
8		ffn 4562	. . . . . . . . 9 |- (F:X-->Y -> F Fn X)
9	7, 8	syl 12	. . . . . . . 8 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> F Fn X)
10	5, 1	grpidcl 9343	. . . . . . . . 9 |- (G e. Grp -> W e. X)
11	10	3ad2ant1 897	. . . . . . . 8 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> W e. X)
12		fnsnfv 4728	. . . . . . . 8 |- ((F Fn X /\ W e. X) -> {(F` W)} = (F"{W}))
13	9, 11, 12	syl11anc 524	. . . . . . 7 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> {(F` W)} = (F"{W}))
14	4, 13	eqtr3d 1927	. . . . . 6 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> {U} = (F"{W}))
15	14	imaeq2d 4264	. . . . 5 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> (`'F"{U}) = (`'F"(F"{W})))
16	15	adantl 424	. . . 4 |- ((F:X-1-1->Y /\ (G e. Grp /\ H e. Grp /\ F e. (G GrpHom H))) -> (`'F"{U}) = (`'F"(F"{W})))
17		f1imacnv 4656	. . . . 5 |- ((F:X-1-1->Y /\ {W} C_ X) -> (`'F"(F"{W})) = {W})
18	10	snssd 3130	. . . . . 6 |- (G e. Grp -> {W} C_ X)
19	18	3ad2ant1 897	. . . . 5 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> {W} C_ X)
20	17, 19	sylan2 500	. . . 4 |- ((F:X-1-1->Y /\ (G e. Grp /\ H e. Grp /\ F e. (G GrpHom H))) -> (`'F"(F"{W})) = {W})
21	16, 20	eqtrd 1925	. . 3 |- ((F:X-1-1->Y /\ (G e. Grp /\ H e. Grp /\ F e. (G GrpHom H))) -> (`'F"{U}) = {W})
22	21	expcom 403	. 2 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> (F:X-1-1->Y -> (`'F"{U}) = {W}))
23		dff13 4850	. . . 4 |- (F:X-1-1->Y <-> (F:X-->Y /\ A.x e. X A.y e. X ((F` x) = (F` y) -> x = y)))
24	7	adantr 425	. . . 4 |- (((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) /\ (`'F"{U}) = {W}) -> F:X-->Y)
25		simpl2 880	. . . . . . . 8 |- (((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) /\ (x e. X /\ y e. X)) -> H e. Grp)
26		ffvelrn 4787	. . . . . . . . . 10 |- ((F:X-->Y /\ x e. X) -> (F` x) e. Y)
27	26, 7	sylan 497	. . . . . . . . 9 |- (((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) /\ x e. X) -> (F` x) e. Y)
28	27	adantrr 431	. . . . . . . 8 |- (((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) /\ (x e. X /\ y e. X)) -> (F` x) e. Y)
29		ffvelrn 4787	. . . . . . . . . 10 |- ((F:X-->Y /\ y e. X) -> (F` y) e. Y)
30	29, 7	sylan 497	. . . . . . . . 9 |- (((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) /\ y e. X) -> (F` y) e. Y)
31	30	adantrl 430	. . . . . . . 8 |- (((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) /\ (x e. X /\ y e. X)) -> (F` y) e. Y)
32		eqid 1884	. . . . . . . . 9 |- ( /g ` H) = ( /g ` H)
33	6, 2, 32	grpeqdivid 16038	. . . . . . . 8 |- ((H e. Grp /\ (F` x) e. Y /\ (F` y) e. Y) -> ((F` x) = (F` y) <-> ((F` x)( /g ` H)(F` y)) = U))
34	25, 28, 31, 33	syl111anc 1100	. . . . . . 7 |- (((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) /\ (x e. X /\ y e. X)) -> ((F` x) = (F` y) <-> ((F` x)( /g ` H)(F` y)) = U))
35	34	adantlr 429	. . . . . 6 |- ((((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) /\ (`'F"{U}) = {W}) /\ (x e. X /\ y e. X)) -> ((F` x) = (F` y) <-> ((F` x)( /g ` H)(F` y)) = U))
36		eqid 1884	. . . . . . . . . 10 |- ( /g ` G) = ( /g ` G)
37	5, 36, 32	ghomdiv 16041	. . . . . . . . 9 |- (((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) /\ (x e. X /\ y e. X)) -> (F` (x( /g ` G)y)) = ((F` x)( /g ` H)(F` y)))
38	37	adantlr 429	. . . . . . . 8 |- ((((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) /\ (`'F"{U}) = {W}) /\ (x e. X /\ y e. X)) -> (F` (x( /g ` G)y)) = ((F` x)( /g ` H)(F` y)))
39	38	eqeq1d 1892	. . . . . . 7 |- ((((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) /\ (`'F"{U}) = {W}) /\ (x e. X /\ y e. X)) -> ((F` (x( /g ` G)y)) = U <-> ((F` x)( /g ` H)(F` y)) = U))
40		ffun 4565	. . . . . . . . . . . . . 14 |- (F:X-->Y -> Fun F)
41	7, 40	syl 12	. . . . . . . . . . . . 13 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> Fun F)
42	41	adantr 425	. . . . . . . . . . . 12 |- (((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) /\ (x e. X /\ y e. X)) -> Fun F)
43	5, 36	grpdivcl 9371	. . . . . . . . . . . . . . 15 |- ((G e. Grp /\ x e. X /\ y e. X) -> (x( /g ` G)y) e. X)
44	43	3expb 1068	. . . . . . . . . . . . . 14 |- ((G e. Grp /\ (x e. X /\ y e. X)) -> (x( /g ` G)y) e. X)
45	44	3ad2antl1 1038	. . . . . . . . . . . . 13 |- (((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) /\ (x e. X /\ y e. X)) -> (x( /g ` G)y) e. X)
46		fdm 4567	. . . . . . . . . . . . . . 15 |- (F:X-->Y -> dom F = X)
47	7, 46	syl 12	. . . . . . . . . . . . . 14 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> dom F = X)
48	47	adantr 425	. . . . . . . . . . . . 13 |- (((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) /\ (x e. X /\ y e. X)) -> dom F = X)
49	45, 48	eleqtrrd 1974	. . . . . . . . . . . 12 |- (((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) /\ (x e. X /\ y e. X)) -> (x( /g ` G)y) e. dom F)
50		fvimacnv 4778	. . . . . . . . . . . 12 |- ((Fun F /\ (x( /g ` G)y) e. dom F) -> ((F` (x( /g ` G)y)) e. {U} <-> (x( /g ` G)y) e. (`'F"{U})))
51	42, 49, 50	syl11anc 524	. . . . . . . . . . 11 |- (((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) /\ (x e. X /\ y e. X)) -> ((F` (x( /g ` G)y)) e. {U} <-> (x( /g ` G)y) e. (`'F"{U})))
52		eleq2 1958	. . . . . . . . . . 11 |- ((`'F"{U}) = {W} -> ((x( /g ` G)y) e. (`'F"{U}) <-> (x( /g ` G)y) e. {W}))
53	51, 52	sylan9bb 599	. . . . . . . . . 10 |- ((((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) /\ (x e. X /\ y e. X)) /\ (`'F"{U}) = {W}) -> ((F` (x( /g ` G)y)) e. {U} <-> (x( /g ` G)y) e. {W}))
54	53	an1rs 547	. . . . . . . . 9 |- ((((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) /\ (`'F"{U}) = {W}) /\ (x e. X /\ y e. X)) -> ((F` (x( /g ` G)y)) e. {U} <-> (x( /g ` G)y) e. {W}))
55	5, 1, 36	grpeqdivid 16038	. . . . . . . . . . . . . 14 |- ((G e. Grp /\ x e. X /\ y e. X) -> (x = y <-> (x( /g ` G)y) = W))
56	55	biimprd 171	. . . . . . . . . . . . 13 |- ((G e. Grp /\ x e. X /\ y e. X) -> ((x( /g ` G)y) = W -> x = y))
57	56	3expb 1068	. . . . . . . . . . . 12 |- ((G e. Grp /\ (x e. X /\ y e. X)) -> ((x( /g ` G)y) = W -> x = y))
58	57	3ad2antl1 1038	. . . . . . . . . . 11 |- (((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) /\ (x e. X /\ y e. X)) -> ((x( /g ` G)y) = W -> x = y))
59		elsni 3066	. . . . . . . . . . 11 |- ((x( /g ` G)y) e. {W} -> (x( /g ` G)y) = W)
60	58, 59	syl5 20	. . . . . . . . . 10 |- (((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) /\ (x e. X /\ y e. X)) -> ((x( /g ` G)y) e. {W} -> x = y))
61	60	adantlr 429	. . . . . . . . 9 |- ((((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) /\ (`'F"{U}) = {W}) /\ (x e. X /\ y e. X)) -> ((x( /g ` G)y) e. {W} -> x = y))
62	54, 61	sylbid 220	. . . . . . . 8 |- ((((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) /\ (`'F"{U}) = {W}) /\ (x e. X /\ y e. X)) -> ((F` (x( /g ` G)y)) e. {U} -> x = y))
63		fvex 4689	. . . . . . . . . . 11 |- (Id` H) e. _V
64	2, 63	eqeltri 1967	. . . . . . . . . 10 |- U e. _V
65	64	snid 3069	. . . . . . . . 9 |- U e. {U}
66		eleq1 1957	. . . . . . . . 9 |- ((F` (x( /g ` G)y)) = U -> ((F` (x( /g ` G)y)) e. {U} <-> U e. {U}))
67	65, 66	mpbiri 211	. . . . . . . 8 |- ((F` (x( /g ` G)y)) = U -> (F` (x( /g ` G)y)) e. {U})
68	62, 67	syl5 20	. . . . . . 7 |- ((((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) /\ (`'F"{U}) = {W}) /\ (x e. X /\ y e. X)) -> ((F` (x( /g ` G)y)) = U -> x = y))
69	39, 68	sylbird 222	. . . . . 6 |- ((((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) /\ (`'F"{U}) = {W}) /\ (x e. X /\ y e. X)) -> (((F` x)( /g ` H)(F` y)) = U -> x = y))
70	35, 69	sylbid 220	. . . . 5 |- ((((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) /\ (`'F"{U}) = {W}) /\ (x e. X /\ y e. X)) -> ((F` x) = (F` y) -> x = y))
71	70	r19.21aivva 15653	. . . 4 |- (((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) /\ (`'F"{U}) = {W}) -> A.x e. X A.y e. X ((F` x) = (F` y) -> x = y))
72	23, 24, 71	sylanbrc 527	. . 3 |- (((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) /\ (`'F"{U}) = {W}) -> F:X-1-1->Y)
73	72	ex 402	. 2 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> ((`'F"{U}) = {W} -> F:X-1-1->Y))
74	22, 73	impbid 574	1 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> (F:X-1-1->Y <-> (`'F"{U}) = {W}))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105  _Vcvv 2292   C_ wss 2593  {csn 3044  `'ccnv 3985  dom cdm 3986  ran crn 3987  "cima 3989  Fun wfun 3992   Fn wfn 3993  -->wf 3994  -1-1->wf1 3995  ` cfv 3998  (class class class)co 4884  Grpcgr 9311  Idcgi 9312   /g cgs 9314   GrpHom cghom 10189

This theorem is referenced by:  rngkerinj 16129

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-1st 5020  df-2nd 5021  df-grp 9316  df-gid 9317  df-ginv 9318  df-gdiv 9319  df-ghom 10190

Copyright terms: Public domain