Theorem grpokerinj 37917

Description: A group homomorphism is injective if and only if its kernel is zero. (Contributed by Jeff Madsen, 16-Jun-2011.)

Hypotheses

Ref	Expression
grpkerinj.1	⊢ 𝑋 = ran 𝐺
grpkerinj.2	⊢ 𝑊 = (GId‘𝐺)
grpkerinj.3	⊢ 𝑌 = ran 𝐻
grpkerinj.4	⊢ 𝑈 = (GId‘𝐻)

Assertion

Ref	Expression
grpokerinj	⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → (𝐹:𝑋–1-1→𝑌 ↔ (◡𝐹 “ {𝑈}) = {𝑊}))

Proof of Theorem grpokerinj

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		grpkerinj.2	. . . . . . . . 9 ⊢ 𝑊 = (GId‘𝐺)
2		grpkerinj.4	. . . . . . . . 9 ⊢ 𝑈 = (GId‘𝐻)
3	1, 2	ghomidOLD 37913	. . . . . . . 8 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → (𝐹‘𝑊) = 𝑈)
4	3	sneqd 4613	. . . . . . 7 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → {(𝐹‘𝑊)} = {𝑈})
5		grpkerinj.1	. . . . . . . . . 10 ⊢ 𝑋 = ran 𝐺
6		grpkerinj.3	. . . . . . . . . 10 ⊢ 𝑌 = ran 𝐻
7	5, 6	ghomf 37914	. . . . . . . . 9 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → 𝐹:𝑋⟶𝑌)
8	7	ffnd 6707	. . . . . . . 8 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → 𝐹 Fn 𝑋)
9	5, 1	grpoidcl 30495	. . . . . . . . 9 ⊢ (𝐺 ∈ GrpOp → 𝑊 ∈ 𝑋)
10	9	3ad2ant1 1133	. . . . . . . 8 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → 𝑊 ∈ 𝑋)
11		fnsnfv 6958	. . . . . . . 8 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑊 ∈ 𝑋) → {(𝐹‘𝑊)} = (𝐹 “ {𝑊}))
12	8, 10, 11	syl2anc 584	. . . . . . 7 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → {(𝐹‘𝑊)} = (𝐹 “ {𝑊}))
13	4, 12	eqtr3d 2772	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → {𝑈} = (𝐹 “ {𝑊}))
14	13	imaeq2d 6047	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → (◡𝐹 “ {𝑈}) = (◡𝐹 “ (𝐹 “ {𝑊})))
15	14	adantl 481	. . . 4 ⊢ ((𝐹:𝑋–1-1→𝑌 ∧ (𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻))) → (◡𝐹 “ {𝑈}) = (◡𝐹 “ (𝐹 “ {𝑊})))
16	9	snssd 4785	. . . . . 6 ⊢ (𝐺 ∈ GrpOp → {𝑊} ⊆ 𝑋)
17	16	3ad2ant1 1133	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → {𝑊} ⊆ 𝑋)
18		f1imacnv 6834	. . . . 5 ⊢ ((𝐹:𝑋–1-1→𝑌 ∧ {𝑊} ⊆ 𝑋) → (◡𝐹 “ (𝐹 “ {𝑊})) = {𝑊})
19	17, 18	sylan2 593	. . . 4 ⊢ ((𝐹:𝑋–1-1→𝑌 ∧ (𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻))) → (◡𝐹 “ (𝐹 “ {𝑊})) = {𝑊})
20	15, 19	eqtrd 2770	. . 3 ⊢ ((𝐹:𝑋–1-1→𝑌 ∧ (𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻))) → (◡𝐹 “ {𝑈}) = {𝑊})
21	20	expcom 413	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → (𝐹:𝑋–1-1→𝑌 → (◡𝐹 “ {𝑈}) = {𝑊}))
22	7	adantr 480	. . . 4 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (◡𝐹 “ {𝑈}) = {𝑊}) → 𝐹:𝑋⟶𝑌)
23		simpl2 1193	. . . . . . . 8 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝐻 ∈ GrpOp)
24	7	ffvelcdmda 7074	. . . . . . . . 9 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ 𝑥 ∈ 𝑋) → (𝐹‘𝑥) ∈ 𝑌)
25	24	adantrr 717	. . . . . . . 8 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝐹‘𝑥) ∈ 𝑌)
26	7	ffvelcdmda 7074	. . . . . . . . 9 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ 𝑦 ∈ 𝑋) → (𝐹‘𝑦) ∈ 𝑌)
27	26	adantrl 716	. . . . . . . 8 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝐹‘𝑦) ∈ 𝑌)
28		eqid 2735	. . . . . . . . 9 ⊢ ( /𝑔 ‘𝐻) = ( /𝑔 ‘𝐻)
29	6, 2, 28	grpoeqdivid 37905	. . . . . . . 8 ⊢ ((𝐻 ∈ GrpOp ∧ (𝐹‘𝑥) ∈ 𝑌 ∧ (𝐹‘𝑦) ∈ 𝑌) → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ ((𝐹‘𝑥)( /𝑔 ‘𝐻)(𝐹‘𝑦)) = 𝑈))
30	23, 25, 27, 29	syl3anc 1373	. . . . . . 7 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ ((𝐹‘𝑥)( /𝑔 ‘𝐻)(𝐹‘𝑦)) = 𝑈))
31	30	adantlr 715	. . . . . 6 ⊢ ((((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (◡𝐹 “ {𝑈}) = {𝑊}) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ ((𝐹‘𝑥)( /𝑔 ‘𝐻)(𝐹‘𝑦)) = 𝑈))
32		eqid 2735	. . . . . . . . . 10 ⊢ ( /𝑔 ‘𝐺) = ( /𝑔 ‘𝐺)
33	5, 32, 28	ghomdiv 37916	. . . . . . . . 9 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝐹‘(𝑥( /𝑔 ‘𝐺)𝑦)) = ((𝐹‘𝑥)( /𝑔 ‘𝐻)(𝐹‘𝑦)))
34	33	adantlr 715	. . . . . . . 8 ⊢ ((((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (◡𝐹 “ {𝑈}) = {𝑊}) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝐹‘(𝑥( /𝑔 ‘𝐺)𝑦)) = ((𝐹‘𝑥)( /𝑔 ‘𝐻)(𝐹‘𝑦)))
35	34	eqeq1d 2737	. . . . . . 7 ⊢ ((((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (◡𝐹 “ {𝑈}) = {𝑊}) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝐹‘(𝑥( /𝑔 ‘𝐺)𝑦)) = 𝑈 ↔ ((𝐹‘𝑥)( /𝑔 ‘𝐻)(𝐹‘𝑦)) = 𝑈))
36	2	fvexi 6890	. . . . . . . . . 10 ⊢ 𝑈 ∈ V
37	36	snid 4638	. . . . . . . . 9 ⊢ 𝑈 ∈ {𝑈}
38		eleq1 2822	. . . . . . . . 9 ⊢ ((𝐹‘(𝑥( /𝑔 ‘𝐺)𝑦)) = 𝑈 → ((𝐹‘(𝑥( /𝑔 ‘𝐺)𝑦)) ∈ {𝑈} ↔ 𝑈 ∈ {𝑈}))
39	37, 38	mpbiri 258	. . . . . . . 8 ⊢ ((𝐹‘(𝑥( /𝑔 ‘𝐺)𝑦)) = 𝑈 → (𝐹‘(𝑥( /𝑔 ‘𝐺)𝑦)) ∈ {𝑈})
40	7	ffund 6710	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → Fun 𝐹)
41	40	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → Fun 𝐹)
42	5, 32	grpodivcl 30520	. . . . . . . . . . . . . . 15 ⊢ ((𝐺 ∈ GrpOp ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥( /𝑔 ‘𝐺)𝑦) ∈ 𝑋)
43	42	3expb 1120	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥( /𝑔 ‘𝐺)𝑦) ∈ 𝑋)
44	43	3ad2antl1 1186	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥( /𝑔 ‘𝐺)𝑦) ∈ 𝑋)
45	7	fdmd 6716	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → dom 𝐹 = 𝑋)
46	45	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → dom 𝐹 = 𝑋)
47	44, 46	eleqtrrd 2837	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥( /𝑔 ‘𝐺)𝑦) ∈ dom 𝐹)
48		fvimacnv 7043	. . . . . . . . . . . 12 ⊢ ((Fun 𝐹 ∧ (𝑥( /𝑔 ‘𝐺)𝑦) ∈ dom 𝐹) → ((𝐹‘(𝑥( /𝑔 ‘𝐺)𝑦)) ∈ {𝑈} ↔ (𝑥( /𝑔 ‘𝐺)𝑦) ∈ (◡𝐹 “ {𝑈})))
49	41, 47, 48	syl2anc 584	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝐹‘(𝑥( /𝑔 ‘𝐺)𝑦)) ∈ {𝑈} ↔ (𝑥( /𝑔 ‘𝐺)𝑦) ∈ (◡𝐹 “ {𝑈})))
50		eleq2 2823	. . . . . . . . . . 11 ⊢ ((◡𝐹 “ {𝑈}) = {𝑊} → ((𝑥( /𝑔 ‘𝐺)𝑦) ∈ (◡𝐹 “ {𝑈}) ↔ (𝑥( /𝑔 ‘𝐺)𝑦) ∈ {𝑊}))
51	49, 50	sylan9bb 509	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ (◡𝐹 “ {𝑈}) = {𝑊}) → ((𝐹‘(𝑥( /𝑔 ‘𝐺)𝑦)) ∈ {𝑈} ↔ (𝑥( /𝑔 ‘𝐺)𝑦) ∈ {𝑊}))
52	51	an32s 652	. . . . . . . . 9 ⊢ ((((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (◡𝐹 “ {𝑈}) = {𝑊}) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝐹‘(𝑥( /𝑔 ‘𝐺)𝑦)) ∈ {𝑈} ↔ (𝑥( /𝑔 ‘𝐺)𝑦) ∈ {𝑊}))
53		elsni 4618	. . . . . . . . . . 11 ⊢ ((𝑥( /𝑔 ‘𝐺)𝑦) ∈ {𝑊} → (𝑥( /𝑔 ‘𝐺)𝑦) = 𝑊)
54	5, 1, 32	grpoeqdivid 37905	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ GrpOp ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥 = 𝑦 ↔ (𝑥( /𝑔 ‘𝐺)𝑦) = 𝑊))
55	54	biimprd 248	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ GrpOp ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((𝑥( /𝑔 ‘𝐺)𝑦) = 𝑊 → 𝑥 = 𝑦))
56	55	3expb 1120	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝑥( /𝑔 ‘𝐺)𝑦) = 𝑊 → 𝑥 = 𝑦))
57	56	3ad2antl1 1186	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝑥( /𝑔 ‘𝐺)𝑦) = 𝑊 → 𝑥 = 𝑦))
58	53, 57	syl5 34	. . . . . . . . . 10 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝑥( /𝑔 ‘𝐺)𝑦) ∈ {𝑊} → 𝑥 = 𝑦))
59	58	adantlr 715	. . . . . . . . 9 ⊢ ((((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (◡𝐹 “ {𝑈}) = {𝑊}) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝑥( /𝑔 ‘𝐺)𝑦) ∈ {𝑊} → 𝑥 = 𝑦))
60	52, 59	sylbid 240	. . . . . . . 8 ⊢ ((((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (◡𝐹 “ {𝑈}) = {𝑊}) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝐹‘(𝑥( /𝑔 ‘𝐺)𝑦)) ∈ {𝑈} → 𝑥 = 𝑦))
61	39, 60	syl5 34	. . . . . . 7 ⊢ ((((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (◡𝐹 “ {𝑈}) = {𝑊}) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝐹‘(𝑥( /𝑔 ‘𝐺)𝑦)) = 𝑈 → 𝑥 = 𝑦))
62	35, 61	sylbird 260	. . . . . 6 ⊢ ((((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (◡𝐹 “ {𝑈}) = {𝑊}) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (((𝐹‘𝑥)( /𝑔 ‘𝐻)(𝐹‘𝑦)) = 𝑈 → 𝑥 = 𝑦))
63	31, 62	sylbid 240	. . . . 5 ⊢ ((((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (◡𝐹 “ {𝑈}) = {𝑊}) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
64	63	ralrimivva 3187	. . . 4 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (◡𝐹 “ {𝑈}) = {𝑊}) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
65		dff13 7247	. . . 4 ⊢ (𝐹:𝑋–1-1→𝑌 ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
66	22, 64, 65	sylanbrc 583	. . 3 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (◡𝐹 “ {𝑈}) = {𝑊}) → 𝐹:𝑋–1-1→𝑌)
67	66	ex 412	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → ((◡𝐹 “ {𝑈}) = {𝑊} → 𝐹:𝑋–1-1→𝑌))
68	21, 67	impbid 212	1 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → (𝐹:𝑋–1-1→𝑌 ↔ (◡𝐹 “ {𝑈}) = {𝑊}))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2108  ∀wral 3051   ⊆ wss 3926  {csn 4601  ^◡ccnv 5653  dom cdm 5654  ran crn 5655   “ cima 5657  Fun wfun 6525   Fn wfn 6526  ⟶wf 6527  –1-1→wf1 6528  ‘cfv 6531  (class class class)co 7405  GrpOpcgr 30470  GIdcgi 30471   /_𝑔 cgs 30473   GrpOpHom cghomOLD 37907

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7988  df-2nd 7989  df-grpo 30474  df-gid 30475  df-ginv 30476  df-gdiv 30477  df-ghomOLD 37908

This theorem is referenced by:  rngokerinj  37999