Theorem grpokerinj 38828

Description: Obsolete theorem, use kerf1ghm 19461 instead. A group homomorphism is injective if and only if its kernel is zero. (Contributed by Jeff Madsen, 16-Jun-2011.) (New usage is discouraged.) (Proof modification is discouraged.)

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 38824	. . . . . . . 8 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → (𝐹‘𝑊) = 𝑈)
4	3	sneqd 4595	. . . . . . 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 38825	. . . . . . . . 9 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → 𝐹:𝑋⟶𝑌)
8	7	ffnd 6710	. . . . . . . 8 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → 𝐹 Fn 𝑋)
9	5, 1	grpoidcl 31116	. . . . . . . . 9 ⊢ (𝐺 ∈ GrpOp → 𝑊 ∈ 𝑋)
10	9	3ad2ant1 1151	. . . . . . . 8 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → 𝑊 ∈ 𝑋)
11		fnsnfv 6964	. . . . . . . 8 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑊 ∈ 𝑋) → {(𝐹‘𝑊)} = (𝐹 “ {𝑊}))
12	8, 10, 11	syl2anc 596	. . . . . . 7 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → {(𝐹‘𝑊)} = (𝐹 “ {𝑊}))
13	4, 12	eqtr3d 2797	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → {𝑈} = (𝐹 “ {𝑊}))
14	13	imaeq2d 6051	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → (◡𝐹 “ {𝑈}) = (◡𝐹 “ (𝐹 “ {𝑊})))
15	14	adantl 487	. . . 4 ⊢ ((𝐹:𝑋–1-1→𝑌 ∧ (𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻))) → (◡𝐹 “ {𝑈}) = (◡𝐹 “ (𝐹 “ {𝑊})))
16	9	snssd 4746	. . . . . 6 ⊢ (𝐺 ∈ GrpOp → {𝑊} ⊆ 𝑋)
17	16	3ad2ant1 1151	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → {𝑊} ⊆ 𝑋)
18		f1imacnv 6841	. . . . 5 ⊢ ((𝐹:𝑋–1-1→𝑌 ∧ {𝑊} ⊆ 𝑋) → (◡𝐹 “ (𝐹 “ {𝑊})) = {𝑊})
19	17, 18	sylan2 605	. . . 4 ⊢ ((𝐹:𝑋–1-1→𝑌 ∧ (𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻))) → (◡𝐹 “ (𝐹 “ {𝑊})) = {𝑊})
20	15, 19	eqtrd 2795	. . 3 ⊢ ((𝐹:𝑋–1-1→𝑌 ∧ (𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻))) → (◡𝐹 “ {𝑈}) = {𝑊})
21	20	expcom 419	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → (𝐹:𝑋–1-1→𝑌 → (◡𝐹 “ {𝑈}) = {𝑊}))
22	7	adantr 486	. . . 4 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (◡𝐹 “ {𝑈}) = {𝑊}) → 𝐹:𝑋⟶𝑌)
23		simpl2 1211	. . . . . . . 8 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝐻 ∈ GrpOp)
24	7	ffvelcdmda 7084	. . . . . . . . 9 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ 𝑥 ∈ 𝑋) → (𝐹‘𝑥) ∈ 𝑌)
25	24	adantrr 730	. . . . . . . 8 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝐹‘𝑥) ∈ 𝑌)
26	7	ffvelcdmda 7084	. . . . . . . . 9 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ 𝑦 ∈ 𝑋) → (𝐹‘𝑦) ∈ 𝑌)
27	26	adantrl 729	. . . . . . . 8 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝐹‘𝑦) ∈ 𝑌)
28		eqid 2760	. . . . . . . . 9 ⊢ ( /𝑔 ‘𝐻) = ( /𝑔 ‘𝐻)
29	6, 2, 28	grpoeqdivid 38816	. . . . . . . 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 1398	. . . . . . 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 728	. . . . . 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 2760	. . . . . . . . . 10 ⊢ ( /𝑔 ‘𝐺) = ( /𝑔 ‘𝐺)
33	5, 32, 28	ghomdiv 38827	. . . . . . . . 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 728	. . . . . . . 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 2762	. . . . . . 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 6899	. . . . . . . . . 10 ⊢ 𝑈 ∈ V
37	36	snid 4622	. . . . . . . . 9 ⊢ 𝑈 ∈ {𝑈}
38		eleq1 2848	. . . . . . . . 9 ⊢ ((𝐹‘(𝑥( /𝑔 ‘𝐺)𝑦)) = 𝑈 → ((𝐹‘(𝑥( /𝑔 ‘𝐺)𝑦)) ∈ {𝑈} ↔ 𝑈 ∈ {𝑈}))
39	37, 38	mpbiri 261	. . . . . . . 8 ⊢ ((𝐹‘(𝑥( /𝑔 ‘𝐺)𝑦)) = 𝑈 → (𝐹‘(𝑥( /𝑔 ‘𝐺)𝑦)) ∈ {𝑈})
40	7	ffund 6714	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → Fun 𝐹)
41	40	adantr 486	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → Fun 𝐹)
42	5, 32	grpodivcl 31141	. . . . . . . . . . . . . . 15 ⊢ ((𝐺 ∈ GrpOp ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥( /𝑔 ‘𝐺)𝑦) ∈ 𝑋)
43	42	3expb 1138	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥( /𝑔 ‘𝐺)𝑦) ∈ 𝑋)
44	43	3ad2antl1 1204	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥( /𝑔 ‘𝐺)𝑦) ∈ 𝑋)
45	7	fdmd 6720	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → dom 𝐹 = 𝑋)
46	45	adantr 486	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → dom 𝐹 = 𝑋)
47	44, 46	eleqtrrd 2863	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥( /𝑔 ‘𝐺)𝑦) ∈ dom 𝐹)
48		fvimacnv 7052	. . . . . . . . . . . 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 596	. . . . . . . . . . 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 2849	. . . . . . . . . . 11 ⊢ ((◡𝐹 “ {𝑈}) = {𝑊} → ((𝑥( /𝑔 ‘𝐺)𝑦) ∈ (◡𝐹 “ {𝑈}) ↔ (𝑥( /𝑔 ‘𝐺)𝑦) ∈ {𝑊}))
51	49, 50	sylan9bb 519	. . . . . . . . . 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 665	. . . . . . . . 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 4600	. . . . . . . . . . 11 ⊢ ((𝑥( /𝑔 ‘𝐺)𝑦) ∈ {𝑊} → (𝑥( /𝑔 ‘𝐺)𝑦) = 𝑊)
54	5, 1, 32	grpoeqdivid 38816	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ GrpOp ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥 = 𝑦 ↔ (𝑥( /𝑔 ‘𝐺)𝑦) = 𝑊))
55	54	biimprd 251	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ GrpOp ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((𝑥( /𝑔 ‘𝐺)𝑦) = 𝑊 → 𝑥 = 𝑦))
56	55	3expb 1138	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝑥( /𝑔 ‘𝐺)𝑦) = 𝑊 → 𝑥 = 𝑦))
57	56	3ad2antl1 1204	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝑥( /𝑔 ‘𝐺)𝑦) = 𝑊 → 𝑥 = 𝑦))
58	53, 57	syl5 35	. . . . . . . . . 10 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝑥( /𝑔 ‘𝐺)𝑦) ∈ {𝑊} → 𝑥 = 𝑦))
59	58	adantlr 728	. . . . . . . . 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 243	. . . . . . . 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 35	. . . . . . 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 263	. . . . . 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 243	. . . . 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 3205	. . . 4 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (◡𝐹 “ {𝑈}) = {𝑊}) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
65		dff13 7258	. . . 4 ⊢ (𝐹:𝑋–1-1→𝑌 ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
66	22, 64, 65	sylanbrc 595	. . 3 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (◡𝐹 “ {𝑈}) = {𝑊}) → 𝐹:𝑋–1-1→𝑌)
67	66	ex 418	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → ((◡𝐹 “ {𝑈}) = {𝑊} → 𝐹:𝑋–1-1→𝑌))
68	21, 67	impbid 215	1 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → (𝐹:𝑋–1-1→𝑌 ↔ (◡𝐹 “ {𝑈}) = {𝑊}))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  ∀wral 3076   ⊆ wss 3895  {csn 4583  ^◡ccnv 5649  dom cdm 5650  ran crn 5651   “ cima 5653  Fun wfun 6532   Fn wfn 6533  ⟶wf 6534  –1-1→wf1 6535  ‘cfv 6538  (class class class)co 7420  GrpOpcgr 31091  GIdcgi 31092   /_𝑔 cgs 31094   GrpOpHom cghomOLD 38818

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 8001  df-2nd 8002  df-grpo 31095  df-gid 31096  df-ginv 31097  df-gdiv 31098  df-ghomOLD 38819

This theorem is used by:  rngokerinj  38910