Theorem grpokerinj 34724

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 34720	. . . . . . . 8 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → (𝐹‘𝑊) = 𝑈)
4	3	sneqd 4490	. . . . . . 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 34721	. . . . . . . . 9 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → 𝐹:𝑋⟶𝑌)
8	7	ffnd 6390	. . . . . . . 8 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → 𝐹 Fn 𝑋)
9	5, 1	grpoidcl 27978	. . . . . . . . 9 ⊢ (𝐺 ∈ GrpOp → 𝑊 ∈ 𝑋)
10	9	3ad2ant1 1126	. . . . . . . 8 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → 𝑊 ∈ 𝑋)
11		fnsnfv 6617	. . . . . . . 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 2835	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → {𝑈} = (𝐹 “ {𝑊}))
14	13	imaeq2d 5813	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → (◡𝐹 “ {𝑈}) = (◡𝐹 “ (𝐹 “ {𝑊})))
15	14	adantl 482	. . . 4 ⊢ ((𝐹:𝑋–1-1→𝑌 ∧ (𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻))) → (◡𝐹 “ {𝑈}) = (◡𝐹 “ (𝐹 “ {𝑊})))
16	9	snssd 4655	. . . . . 6 ⊢ (𝐺 ∈ GrpOp → {𝑊} ⊆ 𝑋)
17	16	3ad2ant1 1126	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → {𝑊} ⊆ 𝑋)
18		f1imacnv 6506	. . . . 5 ⊢ ((𝐹:𝑋–1-1→𝑌 ∧ {𝑊} ⊆ 𝑋) → (◡𝐹 “ (𝐹 “ {𝑊})) = {𝑊})
19	17, 18	sylan2 592	. . . 4 ⊢ ((𝐹:𝑋–1-1→𝑌 ∧ (𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻))) → (◡𝐹 “ (𝐹 “ {𝑊})) = {𝑊})
20	15, 19	eqtrd 2833	. . 3 ⊢ ((𝐹:𝑋–1-1→𝑌 ∧ (𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻))) → (◡𝐹 “ {𝑈}) = {𝑊})
21	20	expcom 414	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → (𝐹:𝑋–1-1→𝑌 → (◡𝐹 “ {𝑈}) = {𝑊}))
22	7	adantr 481	. . . 4 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (◡𝐹 “ {𝑈}) = {𝑊}) → 𝐹:𝑋⟶𝑌)
23		simpl2 1185	. . . . . . . 8 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝐻 ∈ GrpOp)
24	7	ffvelrnda 6723	. . . . . . . . 9 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ 𝑥 ∈ 𝑋) → (𝐹‘𝑥) ∈ 𝑌)
25	24	adantrr 713	. . . . . . . 8 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝐹‘𝑥) ∈ 𝑌)
26	7	ffvelrnda 6723	. . . . . . . . 9 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ 𝑦 ∈ 𝑋) → (𝐹‘𝑦) ∈ 𝑌)
27	26	adantrl 712	. . . . . . . 8 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝐹‘𝑦) ∈ 𝑌)
28		eqid 2797	. . . . . . . . 9 ⊢ ( /𝑔 ‘𝐻) = ( /𝑔 ‘𝐻)
29	6, 2, 28	grpoeqdivid 34712	. . . . . . . 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 1364	. . . . . . 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 711	. . . . . 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 2797	. . . . . . . . . 10 ⊢ ( /𝑔 ‘𝐺) = ( /𝑔 ‘𝐺)
33	5, 32, 28	ghomdiv 34723	. . . . . . . . 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 711	. . . . . . . 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 2799	. . . . . . 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 6559	. . . . . . . . . 10 ⊢ 𝑈 ∈ V
37	36	snid 4512	. . . . . . . . 9 ⊢ 𝑈 ∈ {𝑈}
38		eleq1 2872	. . . . . . . . 9 ⊢ ((𝐹‘(𝑥( /𝑔 ‘𝐺)𝑦)) = 𝑈 → ((𝐹‘(𝑥( /𝑔 ‘𝐺)𝑦)) ∈ {𝑈} ↔ 𝑈 ∈ {𝑈}))
39	37, 38	mpbiri 259	. . . . . . . 8 ⊢ ((𝐹‘(𝑥( /𝑔 ‘𝐺)𝑦)) = 𝑈 → (𝐹‘(𝑥( /𝑔 ‘𝐺)𝑦)) ∈ {𝑈})
40	7	ffund 6393	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → Fun 𝐹)
41	40	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → Fun 𝐹)
42	5, 32	grpodivcl 28003	. . . . . . . . . . . . . . 15 ⊢ ((𝐺 ∈ GrpOp ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥( /𝑔 ‘𝐺)𝑦) ∈ 𝑋)
43	42	3expb 1113	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥( /𝑔 ‘𝐺)𝑦) ∈ 𝑋)
44	43	3ad2antl1 1178	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥( /𝑔 ‘𝐺)𝑦) ∈ 𝑋)
45	7	fdmd 6398	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → dom 𝐹 = 𝑋)
46	45	adantr 481	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → dom 𝐹 = 𝑋)
47	44, 46	eleqtrrd 2888	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥( /𝑔 ‘𝐺)𝑦) ∈ dom 𝐹)
48		fvimacnv 6695	. . . . . . . . . . . 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 2873	. . . . . . . . . . 11 ⊢ ((◡𝐹 “ {𝑈}) = {𝑊} → ((𝑥( /𝑔 ‘𝐺)𝑦) ∈ (◡𝐹 “ {𝑈}) ↔ (𝑥( /𝑔 ‘𝐺)𝑦) ∈ {𝑊}))
51	49, 50	sylan9bb 510	. . . . . . . . . 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 648	. . . . . . . . 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 4495	. . . . . . . . . . 11 ⊢ ((𝑥( /𝑔 ‘𝐺)𝑦) ∈ {𝑊} → (𝑥( /𝑔 ‘𝐺)𝑦) = 𝑊)
54	5, 1, 32	grpoeqdivid 34712	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ GrpOp ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥 = 𝑦 ↔ (𝑥( /𝑔 ‘𝐺)𝑦) = 𝑊))
55	54	biimprd 249	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ GrpOp ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((𝑥( /𝑔 ‘𝐺)𝑦) = 𝑊 → 𝑥 = 𝑦))
56	55	3expb 1113	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝑥( /𝑔 ‘𝐺)𝑦) = 𝑊 → 𝑥 = 𝑦))
57	56	3ad2antl1 1178	. . . . . . . . . . 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 711	. . . . . . . . 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 241	. . . . . . . 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 261	. . . . . 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 241	. . . . 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 3160	. . . 4 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (◡𝐹 “ {𝑈}) = {𝑊}) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
65		dff13 6885	. . . 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 413	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → ((◡𝐹 “ {𝑈}) = {𝑊} → 𝐹:𝑋–1-1→𝑌))
68	21, 67	impbid 213	1 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → (𝐹:𝑋–1-1→𝑌 ↔ (◡𝐹 “ {𝑈}) = {𝑊}))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1080   = wceq 1525   ∈ wcel 2083  ∀wral 3107   ⊆ wss 3865  {csn 4478  ^◡ccnv 5449  dom cdm 5450  ran crn 5451   “ cima 5453  Fun wfun 6226   Fn wfn 6227  ⟶wf 6228  –1-1→wf1 6229  ‘cfv 6232  (class class class)co 7023  GrpOpcgr 27953  GIdcgi 27954   /_𝑔 cgs 27956   GrpOpHom cghomOLD 34714

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-rep 5088  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228  ax-un 7326

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-ral 3112  df-rex 3113  df-reu 3114  df-rab 3116  df-v 3442  df-sbc 3712  df-csb 3818  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-pw 4461  df-sn 4479  df-pr 4481  df-op 4485  df-uni 4752  df-iun 4833  df-br 4969  df-opab 5031  df-mpt 5048  df-id 5355  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-iota 6196  df-fun 6234  df-fn 6235  df-f 6236  df-f1 6237  df-fo 6238  df-f1o 6239  df-fv 6240  df-riota 6984  df-ov 7026  df-oprab 7027  df-mpo 7028  df-1st 7552  df-2nd 7553  df-grpo 27957  df-gid 27958  df-ginv 27959  df-gdiv 27960  df-ghomOLD 34715

This theorem is referenced by:  rngokerinj  34806