Theorem fvineqsneu 34695

Description: A theorem about functions where the image of every point intersects the domain only at that point. (Contributed by ML, 27-Mar-2021.)

Assertion

Ref	Expression
fvineqsneu	⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) → ∀𝑞 ∈ 𝐴 ∃!𝑥 ∈ ran 𝐹 𝑞 ∈ 𝑥)

Distinct variable groups:   𝐴,𝑞,𝑥   𝐹,𝑞,𝑥   𝐴,𝑝   𝐹,𝑝

Proof of Theorem fvineqsneu

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

Step	Hyp	Ref	Expression
1		fnfvelrn 6848	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑜 ∈ 𝐴) → (𝐹‘𝑜) ∈ ran 𝐹)
2	1	ex 415	. . . . 5 ⊢ (𝐹 Fn 𝐴 → (𝑜 ∈ 𝐴 → (𝐹‘𝑜) ∈ ran 𝐹))
3	2	adantr 483	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) → (𝑜 ∈ 𝐴 → (𝐹‘𝑜) ∈ ran 𝐹))
4		fnrnfv 6725	. . . . . . . . . 10 ⊢ (𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑝 ∈ 𝐴 𝑦 = (𝐹‘𝑝)})
5	4	abeq2d 2947	. . . . . . . . 9 ⊢ (𝐹 Fn 𝐴 → (𝑦 ∈ ran 𝐹 ↔ ∃𝑝 ∈ 𝐴 𝑦 = (𝐹‘𝑝)))
6	5	adantr 483	. . . . . . . 8 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) → (𝑦 ∈ ran 𝐹 ↔ ∃𝑝 ∈ 𝐴 𝑦 = (𝐹‘𝑝)))
7		nfv 1915	. . . . . . . . . 10 ⊢ Ⅎ𝑝 𝐹 Fn 𝐴
8		nfra1 3219	. . . . . . . . . 10 ⊢ Ⅎ𝑝∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}
9	7, 8	nfan 1900	. . . . . . . . 9 ⊢ Ⅎ𝑝(𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝})
10		nfv 1915	. . . . . . . . 9 ⊢ Ⅎ𝑝∀𝑜 ∈ 𝐴 (𝑜 ∈ 𝑦 ↔ 𝑦 = (𝐹‘𝑜))
11		eleq2 2901	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = (𝐹‘𝑝) → (𝑜 ∈ 𝑦 ↔ 𝑜 ∈ (𝐹‘𝑝)))
12		elin 4169	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑜 ∈ ((𝐹‘𝑝) ∩ 𝐴) ↔ (𝑜 ∈ (𝐹‘𝑝) ∧ 𝑜 ∈ 𝐴))
13	12	rbaib 541	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑜 ∈ 𝐴 → (𝑜 ∈ ((𝐹‘𝑝) ∩ 𝐴) ↔ 𝑜 ∈ (𝐹‘𝑝)))
14	13	ad2antll 727	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ (𝑝 ∈ 𝐴 ∧ 𝑜 ∈ 𝐴)) → (𝑜 ∈ ((𝐹‘𝑝) ∩ 𝐴) ↔ 𝑜 ∈ (𝐹‘𝑝)))
15		rsp 3205	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝} → (𝑝 ∈ 𝐴 → ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}))
16		eleq2 2901	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝐹‘𝑝) ∩ 𝐴) = {𝑝} → (𝑜 ∈ ((𝐹‘𝑝) ∩ 𝐴) ↔ 𝑜 ∈ {𝑝}))
17		velsn 4583	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑜 ∈ {𝑝} ↔ 𝑜 = 𝑝)
18		equcom 2025	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑜 = 𝑝 ↔ 𝑝 = 𝑜)
19	17, 18	bitri 277	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑜 ∈ {𝑝} ↔ 𝑝 = 𝑜)
20	16, 19	syl6bb 289	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝐹‘𝑝) ∩ 𝐴) = {𝑝} → (𝑜 ∈ ((𝐹‘𝑝) ∩ 𝐴) ↔ 𝑝 = 𝑜))
21	15, 20	syl6 35	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝} → (𝑝 ∈ 𝐴 → (𝑜 ∈ ((𝐹‘𝑝) ∩ 𝐴) ↔ 𝑝 = 𝑜)))
22	21	adantl 484	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) → (𝑝 ∈ 𝐴 → (𝑜 ∈ ((𝐹‘𝑝) ∩ 𝐴) ↔ 𝑝 = 𝑜)))
23	22	adantrd 494	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) → ((𝑝 ∈ 𝐴 ∧ 𝑜 ∈ 𝐴) → (𝑜 ∈ ((𝐹‘𝑝) ∩ 𝐴) ↔ 𝑝 = 𝑜)))
24	23	imp 409	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ (𝑝 ∈ 𝐴 ∧ 𝑜 ∈ 𝐴)) → (𝑜 ∈ ((𝐹‘𝑝) ∩ 𝐴) ↔ 𝑝 = 𝑜))
25	14, 24	bitr3d 283	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ (𝑝 ∈ 𝐴 ∧ 𝑜 ∈ 𝐴)) → (𝑜 ∈ (𝐹‘𝑝) ↔ 𝑝 = 𝑜))
26	11, 25	sylan9bbr 513	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ (𝑝 ∈ 𝐴 ∧ 𝑜 ∈ 𝐴)) ∧ 𝑦 = (𝐹‘𝑝)) → (𝑜 ∈ 𝑦 ↔ 𝑝 = 𝑜))
27	26	ex 415	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ (𝑝 ∈ 𝐴 ∧ 𝑜 ∈ 𝐴)) → (𝑦 = (𝐹‘𝑝) → (𝑜 ∈ 𝑦 ↔ 𝑝 = 𝑜)))
28	27	anass1rs 653	. . . . . . . . . . . . . . 15 ⊢ ((((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑜 ∈ 𝐴) ∧ 𝑝 ∈ 𝐴) → (𝑦 = (𝐹‘𝑝) → (𝑜 ∈ 𝑦 ↔ 𝑝 = 𝑜)))
29	28	impr 457	. . . . . . . . . . . . . 14 ⊢ ((((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑜 ∈ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑝))) → (𝑜 ∈ 𝑦 ↔ 𝑝 = 𝑜))
30	29	an32s 650	. . . . . . . . . . . . 13 ⊢ ((((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ (𝑝 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑝))) ∧ 𝑜 ∈ 𝐴) → (𝑜 ∈ 𝑦 ↔ 𝑝 = 𝑜))
31		eqeq1 2825	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = (𝐹‘𝑝) → (𝑦 = (𝐹‘𝑜) ↔ (𝐹‘𝑝) = (𝐹‘𝑜)))
32		dffn3 6525	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶ran 𝐹)
33		fvineqsnf1 34694	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐹:𝐴⟶ran 𝐹 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) → 𝐹:𝐴–1-1→ran 𝐹)
34	32, 33	sylanb 583	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) → 𝐹:𝐴–1-1→ran 𝐹)
35		dff13 7013	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐹:𝐴–1-1→ran 𝐹 ↔ (𝐹:𝐴⟶ran 𝐹 ∧ ∀𝑝 ∈ 𝐴 ∀𝑜 ∈ 𝐴 ((𝐹‘𝑝) = (𝐹‘𝑜) → 𝑝 = 𝑜)))
36	34, 35	sylib 220	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) → (𝐹:𝐴⟶ran 𝐹 ∧ ∀𝑝 ∈ 𝐴 ∀𝑜 ∈ 𝐴 ((𝐹‘𝑝) = (𝐹‘𝑜) → 𝑝 = 𝑜)))
37	36	simprd 498	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) → ∀𝑝 ∈ 𝐴 ∀𝑜 ∈ 𝐴 ((𝐹‘𝑝) = (𝐹‘𝑜) → 𝑝 = 𝑜))
38		rsp 3205	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (∀𝑝 ∈ 𝐴 ∀𝑜 ∈ 𝐴 ((𝐹‘𝑝) = (𝐹‘𝑜) → 𝑝 = 𝑜) → (𝑝 ∈ 𝐴 → ∀𝑜 ∈ 𝐴 ((𝐹‘𝑝) = (𝐹‘𝑜) → 𝑝 = 𝑜)))
39	37, 38	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) → (𝑝 ∈ 𝐴 → ∀𝑜 ∈ 𝐴 ((𝐹‘𝑝) = (𝐹‘𝑜) → 𝑝 = 𝑜)))
40		rsp 3205	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∀𝑜 ∈ 𝐴 ((𝐹‘𝑝) = (𝐹‘𝑜) → 𝑝 = 𝑜) → (𝑜 ∈ 𝐴 → ((𝐹‘𝑝) = (𝐹‘𝑜) → 𝑝 = 𝑜)))
41	39, 40	syl6 35	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) → (𝑝 ∈ 𝐴 → (𝑜 ∈ 𝐴 → ((𝐹‘𝑝) = (𝐹‘𝑜) → 𝑝 = 𝑜))))
42	41	imp32 421	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ (𝑝 ∈ 𝐴 ∧ 𝑜 ∈ 𝐴)) → ((𝐹‘𝑝) = (𝐹‘𝑜) → 𝑝 = 𝑜))
43		fveq2 6670	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑝 = 𝑜 → (𝐹‘𝑝) = (𝐹‘𝑜))
44	42, 43	impbid1 227	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ (𝑝 ∈ 𝐴 ∧ 𝑜 ∈ 𝐴)) → ((𝐹‘𝑝) = (𝐹‘𝑜) ↔ 𝑝 = 𝑜))
45	31, 44	sylan9bbr 513	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ (𝑝 ∈ 𝐴 ∧ 𝑜 ∈ 𝐴)) ∧ 𝑦 = (𝐹‘𝑝)) → (𝑦 = (𝐹‘𝑜) ↔ 𝑝 = 𝑜))
46	45	ex 415	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ (𝑝 ∈ 𝐴 ∧ 𝑜 ∈ 𝐴)) → (𝑦 = (𝐹‘𝑝) → (𝑦 = (𝐹‘𝑜) ↔ 𝑝 = 𝑜)))
47	46	anass1rs 653	. . . . . . . . . . . . . . 15 ⊢ ((((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑜 ∈ 𝐴) ∧ 𝑝 ∈ 𝐴) → (𝑦 = (𝐹‘𝑝) → (𝑦 = (𝐹‘𝑜) ↔ 𝑝 = 𝑜)))
48	47	impr 457	. . . . . . . . . . . . . 14 ⊢ ((((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑜 ∈ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑝))) → (𝑦 = (𝐹‘𝑜) ↔ 𝑝 = 𝑜))
49	48	an32s 650	. . . . . . . . . . . . 13 ⊢ ((((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ (𝑝 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑝))) ∧ 𝑜 ∈ 𝐴) → (𝑦 = (𝐹‘𝑜) ↔ 𝑝 = 𝑜))
50	30, 49	bitr4d 284	. . . . . . . . . . . 12 ⊢ ((((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ (𝑝 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑝))) ∧ 𝑜 ∈ 𝐴) → (𝑜 ∈ 𝑦 ↔ 𝑦 = (𝐹‘𝑜)))
51	50	ex 415	. . . . . . . . . . 11 ⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ (𝑝 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑝))) → (𝑜 ∈ 𝐴 → (𝑜 ∈ 𝑦 ↔ 𝑦 = (𝐹‘𝑜))))
52	51	ralrimiv 3181	. . . . . . . . . 10 ⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ (𝑝 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑝))) → ∀𝑜 ∈ 𝐴 (𝑜 ∈ 𝑦 ↔ 𝑦 = (𝐹‘𝑜)))
53	52	exp32 423	. . . . . . . . 9 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) → (𝑝 ∈ 𝐴 → (𝑦 = (𝐹‘𝑝) → ∀𝑜 ∈ 𝐴 (𝑜 ∈ 𝑦 ↔ 𝑦 = (𝐹‘𝑜)))))
54	9, 10, 53	rexlimd 3317	. . . . . . . 8 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) → (∃𝑝 ∈ 𝐴 𝑦 = (𝐹‘𝑝) → ∀𝑜 ∈ 𝐴 (𝑜 ∈ 𝑦 ↔ 𝑦 = (𝐹‘𝑜))))
55	6, 54	sylbid 242	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) → (𝑦 ∈ ran 𝐹 → ∀𝑜 ∈ 𝐴 (𝑜 ∈ 𝑦 ↔ 𝑦 = (𝐹‘𝑜))))
56		rsp 3205	. . . . . . 7 ⊢ (∀𝑜 ∈ 𝐴 (𝑜 ∈ 𝑦 ↔ 𝑦 = (𝐹‘𝑜)) → (𝑜 ∈ 𝐴 → (𝑜 ∈ 𝑦 ↔ 𝑦 = (𝐹‘𝑜))))
57	55, 56	syl6 35	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) → (𝑦 ∈ ran 𝐹 → (𝑜 ∈ 𝐴 → (𝑜 ∈ 𝑦 ↔ 𝑦 = (𝐹‘𝑜)))))
58	57	com23 86	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) → (𝑜 ∈ 𝐴 → (𝑦 ∈ ran 𝐹 → (𝑜 ∈ 𝑦 ↔ 𝑦 = (𝐹‘𝑜)))))
59	58	ralrimdv 3188	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) → (𝑜 ∈ 𝐴 → ∀𝑦 ∈ ran 𝐹(𝑜 ∈ 𝑦 ↔ 𝑦 = (𝐹‘𝑜))))
60		reu6i 3719	. . . . 5 ⊢ (((𝐹‘𝑜) ∈ ran 𝐹 ∧ ∀𝑦 ∈ ran 𝐹(𝑜 ∈ 𝑦 ↔ 𝑦 = (𝐹‘𝑜))) → ∃!𝑦 ∈ ran 𝐹 𝑜 ∈ 𝑦)
61	60	ex 415	. . . 4 ⊢ ((𝐹‘𝑜) ∈ ran 𝐹 → (∀𝑦 ∈ ran 𝐹(𝑜 ∈ 𝑦 ↔ 𝑦 = (𝐹‘𝑜)) → ∃!𝑦 ∈ ran 𝐹 𝑜 ∈ 𝑦))
62	3, 59, 61	syl6c 70	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) → (𝑜 ∈ 𝐴 → ∃!𝑦 ∈ ran 𝐹 𝑜 ∈ 𝑦))
63	62	ralrimiv 3181	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) → ∀𝑜 ∈ 𝐴 ∃!𝑦 ∈ ran 𝐹 𝑜 ∈ 𝑦)
64		nfv 1915	. . . 4 ⊢ Ⅎ𝑥 𝑞 = 𝑜
65		nfv 1915	. . . 4 ⊢ Ⅎ𝑦 𝑞 = 𝑜
66		nfvd 1916	. . . 4 ⊢ (𝑞 = 𝑜 → Ⅎ𝑦 𝑞 ∈ 𝑥)
67		nfvd 1916	. . . 4 ⊢ (𝑞 = 𝑜 → Ⅎ𝑥 𝑜 ∈ 𝑦)
68		eleq12 2902	. . . . 5 ⊢ ((𝑞 = 𝑜 ∧ 𝑥 = 𝑦) → (𝑞 ∈ 𝑥 ↔ 𝑜 ∈ 𝑦))
69	68	ex 415	. . . 4 ⊢ (𝑞 = 𝑜 → (𝑥 = 𝑦 → (𝑞 ∈ 𝑥 ↔ 𝑜 ∈ 𝑦)))
70	64, 65, 66, 67, 69	cbvreud 34657	. . 3 ⊢ (𝑞 = 𝑜 → (∃!𝑥 ∈ ran 𝐹 𝑞 ∈ 𝑥 ↔ ∃!𝑦 ∈ ran 𝐹 𝑜 ∈ 𝑦))
71	70	cbvralvw 3449	. 2 ⊢ (∀𝑞 ∈ 𝐴 ∃!𝑥 ∈ ran 𝐹 𝑞 ∈ 𝑥 ↔ ∀𝑜 ∈ 𝐴 ∃!𝑦 ∈ ran 𝐹 𝑜 ∈ 𝑦)
72	63, 71	sylibr 236	1 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) → ∀𝑞 ∈ 𝐴 ∃!𝑥 ∈ ran 𝐹 𝑞 ∈ 𝑥)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∀wral 3138  ∃wrex 3139  ∃!wreu 3140   ∩ cin 3935  {csn 4567  ran crn 5556   Fn wfn 6350  ⟶wf 6351  –1-1→wf1 6352  ‘cfv 6355

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fv 6363

This theorem is referenced by:  fvineqsneq  34696