Theorem fvineqsneu 37434

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 7075	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑜 ∈ 𝐴) → (𝐹‘𝑜) ∈ ran 𝐹)
2	1	ex 412	. . . . 5 ⊢ (𝐹 Fn 𝐴 → (𝑜 ∈ 𝐴 → (𝐹‘𝑜) ∈ ran 𝐹))
3	2	adantr 480	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) → (𝑜 ∈ 𝐴 → (𝐹‘𝑜) ∈ ran 𝐹))
4		fnrnfv 6943	. . . . . . . . . 10 ⊢ (𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑝 ∈ 𝐴 𝑦 = (𝐹‘𝑝)})
5	4	eqabrd 2878	. . . . . . . . 9 ⊢ (𝐹 Fn 𝐴 → (𝑦 ∈ ran 𝐹 ↔ ∃𝑝 ∈ 𝐴 𝑦 = (𝐹‘𝑝)))
6	5	adantr 480	. . . . . . . 8 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) → (𝑦 ∈ ran 𝐹 ↔ ∃𝑝 ∈ 𝐴 𝑦 = (𝐹‘𝑝)))
7		nfv 1914	. . . . . . . . . 10 ⊢ Ⅎ𝑝 𝐹 Fn 𝐴
8		nfra1 3270	. . . . . . . . . 10 ⊢ Ⅎ𝑝∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}
9	7, 8	nfan 1899	. . . . . . . . 9 ⊢ Ⅎ𝑝(𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝})
10		nfv 1914	. . . . . . . . 9 ⊢ Ⅎ𝑝∀𝑜 ∈ 𝐴 (𝑜 ∈ 𝑦 ↔ 𝑦 = (𝐹‘𝑜))
11		eleq2w2 2732	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = (𝐹‘𝑝) → (𝑜 ∈ 𝑦 ↔ 𝑜 ∈ (𝐹‘𝑝)))
12		elin 3947	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑜 ∈ ((𝐹‘𝑝) ∩ 𝐴) ↔ (𝑜 ∈ (𝐹‘𝑝) ∧ 𝑜 ∈ 𝐴))
13	12	rbaib 538	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑜 ∈ 𝐴 → (𝑜 ∈ ((𝐹‘𝑝) ∩ 𝐴) ↔ 𝑜 ∈ (𝐹‘𝑝)))
14	13	ad2antll 729	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ (𝑝 ∈ 𝐴 ∧ 𝑜 ∈ 𝐴)) → (𝑜 ∈ ((𝐹‘𝑝) ∩ 𝐴) ↔ 𝑜 ∈ (𝐹‘𝑝)))
15		rsp 3234	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝} → (𝑝 ∈ 𝐴 → ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}))
16		eleq2w2 2732	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝐹‘𝑝) ∩ 𝐴) = {𝑝} → (𝑜 ∈ ((𝐹‘𝑝) ∩ 𝐴) ↔ 𝑜 ∈ {𝑝}))
17		velsn 4622	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑜 ∈ {𝑝} ↔ 𝑜 = 𝑝)
18		equcom 2018	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑜 = 𝑝 ↔ 𝑝 = 𝑜)
19	17, 18	bitri 275	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑜 ∈ {𝑝} ↔ 𝑝 = 𝑜)
20	16, 19	bitrdi 287	. . . . . . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) → (𝑝 ∈ 𝐴 → (𝑜 ∈ ((𝐹‘𝑝) ∩ 𝐴) ↔ 𝑝 = 𝑜)))
23	22	adantrd 491	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) → ((𝑝 ∈ 𝐴 ∧ 𝑜 ∈ 𝐴) → (𝑜 ∈ ((𝐹‘𝑝) ∩ 𝐴) ↔ 𝑝 = 𝑜)))
24	23	imp 406	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ (𝑝 ∈ 𝐴 ∧ 𝑜 ∈ 𝐴)) → (𝑜 ∈ ((𝐹‘𝑝) ∩ 𝐴) ↔ 𝑝 = 𝑜))
25	14, 24	bitr3d 281	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ (𝑝 ∈ 𝐴 ∧ 𝑜 ∈ 𝐴)) → (𝑜 ∈ (𝐹‘𝑝) ↔ 𝑝 = 𝑜))
26	11, 25	sylan9bbr 510	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ (𝑝 ∈ 𝐴 ∧ 𝑜 ∈ 𝐴)) ∧ 𝑦 = (𝐹‘𝑝)) → (𝑜 ∈ 𝑦 ↔ 𝑝 = 𝑜))
27	26	ex 412	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ (𝑝 ∈ 𝐴 ∧ 𝑜 ∈ 𝐴)) → (𝑦 = (𝐹‘𝑝) → (𝑜 ∈ 𝑦 ↔ 𝑝 = 𝑜)))
28	27	anass1rs 655	. . . . . . . . . . . . . . 15 ⊢ ((((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑜 ∈ 𝐴) ∧ 𝑝 ∈ 𝐴) → (𝑦 = (𝐹‘𝑝) → (𝑜 ∈ 𝑦 ↔ 𝑝 = 𝑜)))
29	28	impr 454	. . . . . . . . . . . . . 14 ⊢ ((((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑜 ∈ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑝))) → (𝑜 ∈ 𝑦 ↔ 𝑝 = 𝑜))
30	29	an32s 652	. . . . . . . . . . . . 13 ⊢ ((((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ (𝑝 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑝))) ∧ 𝑜 ∈ 𝐴) → (𝑜 ∈ 𝑦 ↔ 𝑝 = 𝑜))
31		eqeq1 2740	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = (𝐹‘𝑝) → (𝑦 = (𝐹‘𝑜) ↔ (𝐹‘𝑝) = (𝐹‘𝑜)))
32		dffn3 6723	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶ran 𝐹)
33		fvineqsnf1 37433	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐹:𝐴⟶ran 𝐹 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) → 𝐹:𝐴–1-1→ran 𝐹)
34	32, 33	sylanb 581	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) → 𝐹:𝐴–1-1→ran 𝐹)
35		dff13 7252	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐹:𝐴–1-1→ran 𝐹 ↔ (𝐹:𝐴⟶ran 𝐹 ∧ ∀𝑝 ∈ 𝐴 ∀𝑜 ∈ 𝐴 ((𝐹‘𝑝) = (𝐹‘𝑜) → 𝑝 = 𝑜)))
36	34, 35	sylib 218	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) → (𝐹:𝐴⟶ran 𝐹 ∧ ∀𝑝 ∈ 𝐴 ∀𝑜 ∈ 𝐴 ((𝐹‘𝑝) = (𝐹‘𝑜) → 𝑝 = 𝑜)))
37		rsp 3234	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (∀𝑝 ∈ 𝐴 ∀𝑜 ∈ 𝐴 ((𝐹‘𝑝) = (𝐹‘𝑜) → 𝑝 = 𝑜) → (𝑝 ∈ 𝐴 → ∀𝑜 ∈ 𝐴 ((𝐹‘𝑝) = (𝐹‘𝑜) → 𝑝 = 𝑜)))
38	36, 37	simpl2im 503	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) → (𝑝 ∈ 𝐴 → ∀𝑜 ∈ 𝐴 ((𝐹‘𝑝) = (𝐹‘𝑜) → 𝑝 = 𝑜)))
39		rsp 3234	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∀𝑜 ∈ 𝐴 ((𝐹‘𝑝) = (𝐹‘𝑜) → 𝑝 = 𝑜) → (𝑜 ∈ 𝐴 → ((𝐹‘𝑝) = (𝐹‘𝑜) → 𝑝 = 𝑜)))
40	38, 39	syl6 35	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) → (𝑝 ∈ 𝐴 → (𝑜 ∈ 𝐴 → ((𝐹‘𝑝) = (𝐹‘𝑜) → 𝑝 = 𝑜))))
41	40	imp32 418	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ (𝑝 ∈ 𝐴 ∧ 𝑜 ∈ 𝐴)) → ((𝐹‘𝑝) = (𝐹‘𝑜) → 𝑝 = 𝑜))
42		fveq2 6881	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑝 = 𝑜 → (𝐹‘𝑝) = (𝐹‘𝑜))
43	41, 42	impbid1 225	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ (𝑝 ∈ 𝐴 ∧ 𝑜 ∈ 𝐴)) → ((𝐹‘𝑝) = (𝐹‘𝑜) ↔ 𝑝 = 𝑜))
44	31, 43	sylan9bbr 510	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ (𝑝 ∈ 𝐴 ∧ 𝑜 ∈ 𝐴)) ∧ 𝑦 = (𝐹‘𝑝)) → (𝑦 = (𝐹‘𝑜) ↔ 𝑝 = 𝑜))
45	44	ex 412	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ (𝑝 ∈ 𝐴 ∧ 𝑜 ∈ 𝐴)) → (𝑦 = (𝐹‘𝑝) → (𝑦 = (𝐹‘𝑜) ↔ 𝑝 = 𝑜)))
46	45	anass1rs 655	. . . . . . . . . . . . . . 15 ⊢ ((((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑜 ∈ 𝐴) ∧ 𝑝 ∈ 𝐴) → (𝑦 = (𝐹‘𝑝) → (𝑦 = (𝐹‘𝑜) ↔ 𝑝 = 𝑜)))
47	46	impr 454	. . . . . . . . . . . . . 14 ⊢ ((((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑜 ∈ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑝))) → (𝑦 = (𝐹‘𝑜) ↔ 𝑝 = 𝑜))
48	47	an32s 652	. . . . . . . . . . . . 13 ⊢ ((((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ (𝑝 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑝))) ∧ 𝑜 ∈ 𝐴) → (𝑦 = (𝐹‘𝑜) ↔ 𝑝 = 𝑜))
49	30, 48	bitr4d 282	. . . . . . . . . . . 12 ⊢ ((((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ (𝑝 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑝))) ∧ 𝑜 ∈ 𝐴) → (𝑜 ∈ 𝑦 ↔ 𝑦 = (𝐹‘𝑜)))
50	49	ex 412	. . . . . . . . . . 11 ⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ (𝑝 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑝))) → (𝑜 ∈ 𝐴 → (𝑜 ∈ 𝑦 ↔ 𝑦 = (𝐹‘𝑜))))
51	50	ralrimiv 3132	. . . . . . . . . 10 ⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ (𝑝 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑝))) → ∀𝑜 ∈ 𝐴 (𝑜 ∈ 𝑦 ↔ 𝑦 = (𝐹‘𝑜)))
52	51	exp32 420	. . . . . . . . 9 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) → (𝑝 ∈ 𝐴 → (𝑦 = (𝐹‘𝑝) → ∀𝑜 ∈ 𝐴 (𝑜 ∈ 𝑦 ↔ 𝑦 = (𝐹‘𝑜)))))
53	9, 10, 52	rexlimd 3253	. . . . . . . 8 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) → (∃𝑝 ∈ 𝐴 𝑦 = (𝐹‘𝑝) → ∀𝑜 ∈ 𝐴 (𝑜 ∈ 𝑦 ↔ 𝑦 = (𝐹‘𝑜))))
54	6, 53	sylbid 240	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) → (𝑦 ∈ ran 𝐹 → ∀𝑜 ∈ 𝐴 (𝑜 ∈ 𝑦 ↔ 𝑦 = (𝐹‘𝑜))))
55		rsp 3234	. . . . . . 7 ⊢ (∀𝑜 ∈ 𝐴 (𝑜 ∈ 𝑦 ↔ 𝑦 = (𝐹‘𝑜)) → (𝑜 ∈ 𝐴 → (𝑜 ∈ 𝑦 ↔ 𝑦 = (𝐹‘𝑜))))
56	54, 55	syl6 35	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) → (𝑦 ∈ ran 𝐹 → (𝑜 ∈ 𝐴 → (𝑜 ∈ 𝑦 ↔ 𝑦 = (𝐹‘𝑜)))))
57	56	com23 86	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) → (𝑜 ∈ 𝐴 → (𝑦 ∈ ran 𝐹 → (𝑜 ∈ 𝑦 ↔ 𝑦 = (𝐹‘𝑜)))))
58	57	ralrimdv 3139	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) → (𝑜 ∈ 𝐴 → ∀𝑦 ∈ ran 𝐹(𝑜 ∈ 𝑦 ↔ 𝑦 = (𝐹‘𝑜))))
59		reu6i 3716	. . . . 5 ⊢ (((𝐹‘𝑜) ∈ ran 𝐹 ∧ ∀𝑦 ∈ ran 𝐹(𝑜 ∈ 𝑦 ↔ 𝑦 = (𝐹‘𝑜))) → ∃!𝑦 ∈ ran 𝐹 𝑜 ∈ 𝑦)
60	59	ex 412	. . . 4 ⊢ ((𝐹‘𝑜) ∈ ran 𝐹 → (∀𝑦 ∈ ran 𝐹(𝑜 ∈ 𝑦 ↔ 𝑦 = (𝐹‘𝑜)) → ∃!𝑦 ∈ ran 𝐹 𝑜 ∈ 𝑦))
61	3, 58, 60	syl6c 70	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) → (𝑜 ∈ 𝐴 → ∃!𝑦 ∈ ran 𝐹 𝑜 ∈ 𝑦))
62	61	ralrimiv 3132	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) → ∀𝑜 ∈ 𝐴 ∃!𝑦 ∈ ran 𝐹 𝑜 ∈ 𝑦)
63		nfv 1914	. . . 4 ⊢ Ⅎ𝑥 𝑞 = 𝑜
64		nfv 1914	. . . 4 ⊢ Ⅎ𝑦 𝑞 = 𝑜
65		nfvd 1915	. . . 4 ⊢ (𝑞 = 𝑜 → Ⅎ𝑦 𝑞 ∈ 𝑥)
66		nfvd 1915	. . . 4 ⊢ (𝑞 = 𝑜 → Ⅎ𝑥 𝑜 ∈ 𝑦)
67		elequ12 2127	. . . . 5 ⊢ ((𝑞 = 𝑜 ∧ 𝑥 = 𝑦) → (𝑞 ∈ 𝑥 ↔ 𝑜 ∈ 𝑦))
68	67	ex 412	. . . 4 ⊢ (𝑞 = 𝑜 → (𝑥 = 𝑦 → (𝑞 ∈ 𝑥 ↔ 𝑜 ∈ 𝑦)))
69	63, 64, 65, 66, 68	cbvreud 37396	. . 3 ⊢ (𝑞 = 𝑜 → (∃!𝑥 ∈ ran 𝐹 𝑞 ∈ 𝑥 ↔ ∃!𝑦 ∈ ran 𝐹 𝑜 ∈ 𝑦))
70	69	cbvralvw 3224	. 2 ⊢ (∀𝑞 ∈ 𝐴 ∃!𝑥 ∈ ran 𝐹 𝑞 ∈ 𝑥 ↔ ∀𝑜 ∈ 𝐴 ∃!𝑦 ∈ ran 𝐹 𝑜 ∈ 𝑦)
71	62, 70	sylibr 234	1 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) → ∀𝑞 ∈ 𝐴 ∃!𝑥 ∈ ran 𝐹 𝑞 ∈ 𝑥)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3052  ∃wrex 3061  ∃!wreu 3362   ∩ cin 3930  {csn 4606  ran crn 5660   Fn wfn 6531  ⟶wf 6532  –1-1→wf1 6533  ‘cfv 6536

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

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 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3365  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fv 6544

This theorem is referenced by:  fvineqsneq  37435