Theorem cnextfvval 21679

Description: The value of the continuous extension of a given function 𝐹 at a point 𝑋. (Contributed by Thierry Arnoux, 21-Dec-2017.)

Hypotheses

Ref	Expression
cnextf.1	⊢ 𝐶 = ∪ 𝐽
cnextf.2	⊢ 𝐵 = ∪ 𝐾
cnextf.3	⊢ (𝜑 → 𝐽 ∈ Top)
cnextf.4	⊢ (𝜑 → 𝐾 ∈ Haus)
cnextf.5	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
cnextf.a	⊢ (𝜑 → 𝐴 ⊆ 𝐶)
cnextf.6	⊢ (𝜑 → ((cls‘𝐽)‘𝐴) = 𝐶)
cnextf.7	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) ≠ ∅)

Assertion

Ref	Expression
cnextfvval	⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (((𝐽CnExt𝐾)‘𝐹)‘𝑋) = ∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶   𝑥,𝐹   𝑥,𝐽   𝑥,𝐾   𝑥,𝑋   𝜑,𝑥

Proof of Theorem cnextfvval

Step	Hyp	Ref	Expression
1		cnextf.3	. . . 4 ⊢ (𝜑 → 𝐽 ∈ Top)
2	1	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → 𝐽 ∈ Top)
3		cnextf.4	. . . 4 ⊢ (𝜑 → 𝐾 ∈ Haus)
4	3	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → 𝐾 ∈ Haus)
5		cnextf.5	. . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
6	5	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → 𝐹:𝐴⟶𝐵)
7		cnextf.a	. . . 4 ⊢ (𝜑 → 𝐴 ⊆ 𝐶)
8	7	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → 𝐴 ⊆ 𝐶)
9		cnextf.1	. . . 4 ⊢ 𝐶 = ∪ 𝐽
10		cnextf.2	. . . 4 ⊢ 𝐵 = ∪ 𝐾
11	9, 10	cnextfun 21678	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Haus) ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ 𝐶)) → Fun ((𝐽CnExt𝐾)‘𝐹))
12	2, 4, 6, 8, 11	syl22anc 1319	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → Fun ((𝐽CnExt𝐾)‘𝐹))
13		cnextf.6	. . . . . 6 ⊢ (𝜑 → ((cls‘𝐽)‘𝐴) = 𝐶)
14	13	eleq2d 2673	. . . . 5 ⊢ (𝜑 → (𝑋 ∈ ((cls‘𝐽)‘𝐴) ↔ 𝑋 ∈ 𝐶))
15	14	biimpar 501	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → 𝑋 ∈ ((cls‘𝐽)‘𝐴))
16		fvex 6113	. . . . . . 7 ⊢ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹) ∈ V
17	16	uniex 6851	. . . . . 6 ⊢ ∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹) ∈ V
18	17	snid 4155	. . . . 5 ⊢ ∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹) ∈ {∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹)}
19		sneq 4135	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑋 → {𝑥} = {𝑋})
20	19	fveq2d 6107	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑋 → ((nei‘𝐽)‘{𝑥}) = ((nei‘𝐽)‘{𝑋}))
21	20	oveq1d 6564	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑋 → (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) = (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))
22	21	oveq2d 6565	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → (𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴)) = (𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴)))
23	22	fveq1d 6105	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) = ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹))
24	23	breq1d 4593	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) ≈ 1𝑜 ↔ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹) ≈ 1𝑜))
25	24	imbi2d 329	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → ((𝜑 → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) ≈ 1𝑜) ↔ (𝜑 → ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹) ≈ 1𝑜)))
26	3	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐾 ∈ Haus)
27	1	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐽 ∈ Top)
28	9	toptopon 20548	. . . . . . . . . . . 12 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝐶))
29	27, 28	sylib 207	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐽 ∈ (TopOn‘𝐶))
30	7	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐴 ⊆ 𝐶)
31		simpr 476	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝑥 ∈ 𝐶)
32	13	eleq2d 2673	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↔ 𝑥 ∈ 𝐶))
33	32	biimpar 501	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝑥 ∈ ((cls‘𝐽)‘𝐴))
34		trnei 21506	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ (TopOn‘𝐶) ∧ 𝐴 ⊆ 𝐶 ∧ 𝑥 ∈ 𝐶) → (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↔ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (Fil‘𝐴)))
35	34	biimpa 500	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ (TopOn‘𝐶) ∧ 𝐴 ⊆ 𝐶 ∧ 𝑥 ∈ 𝐶) ∧ 𝑥 ∈ ((cls‘𝐽)‘𝐴)) → (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (Fil‘𝐴))
36	29, 30, 31, 33, 35	syl31anc 1321	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (Fil‘𝐴))
37	5	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐹:𝐴⟶𝐵)
38		cnextf.7	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) ≠ ∅)
39	10	hausflf2 21612	. . . . . . . . . 10 ⊢ (((𝐾 ∈ Haus ∧ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (Fil‘𝐴) ∧ 𝐹:𝐴⟶𝐵) ∧ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) ≠ ∅) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) ≈ 1𝑜)
40	26, 36, 37, 38, 39	syl31anc 1321	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) ≈ 1𝑜)
41	40	expcom 450	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐶 → (𝜑 → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) ≈ 1𝑜))
42	25, 41	vtoclga 3245	. . . . . . 7 ⊢ (𝑋 ∈ 𝐶 → (𝜑 → ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹) ≈ 1𝑜))
43	42	impcom 445	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹) ≈ 1𝑜)
44		en1b 7910	. . . . . 6 ⊢ (((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹) ≈ 1𝑜 ↔ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹) = {∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹)})
45	43, 44	sylib 207	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹) = {∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹)})
46	18, 45	syl5eleqr 2695	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹) ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹))
47		nfiu1 4486	. . . . . . . 8 ⊢ Ⅎ𝑥∪ 𝑥 ∈ ((cls‘𝐽)‘𝐴)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹))
48	47	nfel2 2767	. . . . . . 7 ⊢ Ⅎ𝑥⟨𝑋, ∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹)⟩ ∈ ∪ 𝑥 ∈ ((cls‘𝐽)‘𝐴)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹))
49		nfv 1830	. . . . . . 7 ⊢ Ⅎ𝑥(𝑋 ∈ ((cls‘𝐽)‘𝐴) ∧ ∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹) ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹))
50	48, 49	nfbi 1821	. . . . . 6 ⊢ Ⅎ𝑥(⟨𝑋, ∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹)⟩ ∈ ∪ 𝑥 ∈ ((cls‘𝐽)‘𝐴)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)) ↔ (𝑋 ∈ ((cls‘𝐽)‘𝐴) ∧ ∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹) ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹)))
51		opeq1 4340	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → ⟨𝑥, ∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹)⟩ = ⟨𝑋, ∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹)⟩)
52	51	eleq1d 2672	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (⟨𝑥, ∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹)⟩ ∈ ∪ 𝑥 ∈ ((cls‘𝐽)‘𝐴)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)) ↔ ⟨𝑋, ∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹)⟩ ∈ ∪ 𝑥 ∈ ((cls‘𝐽)‘𝐴)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹))))
53		eleq1 2676	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↔ 𝑋 ∈ ((cls‘𝐽)‘𝐴)))
54	23	eleq2d 2673	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹) ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) ↔ ∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹) ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹)))
55	53, 54	anbi12d 743	. . . . . . 7 ⊢ (𝑥 = 𝑋 → ((𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ ∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹) ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)) ↔ (𝑋 ∈ ((cls‘𝐽)‘𝐴) ∧ ∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹) ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹))))
56	52, 55	bibi12d 334	. . . . . 6 ⊢ (𝑥 = 𝑋 → ((⟨𝑥, ∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹)⟩ ∈ ∪ 𝑥 ∈ ((cls‘𝐽)‘𝐴)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)) ↔ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ ∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹) ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹))) ↔ (⟨𝑋, ∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹)⟩ ∈ ∪ 𝑥 ∈ ((cls‘𝐽)‘𝐴)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)) ↔ (𝑋 ∈ ((cls‘𝐽)‘𝐴) ∧ ∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹) ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹)))))
57		opeliunxp 5093	. . . . . 6 ⊢ (⟨𝑥, ∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹)⟩ ∈ ∪ 𝑥 ∈ ((cls‘𝐽)‘𝐴)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)) ↔ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ ∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹) ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)))
58	50, 56, 57	vtoclg1f 3238	. . . . 5 ⊢ (𝑋 ∈ 𝐶 → (⟨𝑋, ∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹)⟩ ∈ ∪ 𝑥 ∈ ((cls‘𝐽)‘𝐴)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)) ↔ (𝑋 ∈ ((cls‘𝐽)‘𝐴) ∧ ∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹) ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹))))
59	58	adantl 481	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (⟨𝑋, ∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹)⟩ ∈ ∪ 𝑥 ∈ ((cls‘𝐽)‘𝐴)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)) ↔ (𝑋 ∈ ((cls‘𝐽)‘𝐴) ∧ ∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹) ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹))))
60	15, 46, 59	mpbir2and 959	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ⟨𝑋, ∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹)⟩ ∈ ∪ 𝑥 ∈ ((cls‘𝐽)‘𝐴)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)))
61		df-br 4584	. . . 4 ⊢ (𝑋((𝐽CnExt𝐾)‘𝐹)∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹) ↔ ⟨𝑋, ∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹)⟩ ∈ ((𝐽CnExt𝐾)‘𝐹))
62		haustop 20945	. . . . . . . 8 ⊢ (𝐾 ∈ Haus → 𝐾 ∈ Top)
63	3, 62	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ Top)
64	63	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → 𝐾 ∈ Top)
65	9, 10	cnextfval 21676	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ 𝐶)) → ((𝐽CnExt𝐾)‘𝐹) = ∪ 𝑥 ∈ ((cls‘𝐽)‘𝐴)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)))
66	2, 64, 6, 8, 65	syl22anc 1319	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ((𝐽CnExt𝐾)‘𝐹) = ∪ 𝑥 ∈ ((cls‘𝐽)‘𝐴)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)))
67	66	eleq2d 2673	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (⟨𝑋, ∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹)⟩ ∈ ((𝐽CnExt𝐾)‘𝐹) ↔ ⟨𝑋, ∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹)⟩ ∈ ∪ 𝑥 ∈ ((cls‘𝐽)‘𝐴)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹))))
68	61, 67	syl5bb 271	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝑋((𝐽CnExt𝐾)‘𝐹)∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹) ↔ ⟨𝑋, ∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹)⟩ ∈ ∪ 𝑥 ∈ ((cls‘𝐽)‘𝐴)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹))))
69	60, 68	mpbird 246	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → 𝑋((𝐽CnExt𝐾)‘𝐹)∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹))
70		funbrfv 6144	. 2 ⊢ (Fun ((𝐽CnExt𝐾)‘𝐹) → (𝑋((𝐽CnExt𝐾)‘𝐹)∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹) → (((𝐽CnExt𝐾)‘𝐹)‘𝑋) = ∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹)))
71	12, 69, 70	sylc 63	1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (((𝐽CnExt𝐾)‘𝐹)‘𝑋) = ∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780   ⊆ wss 3540  ∅c0 3874  {csn 4125  ⟨cop 4131  ∪ cuni 4372  ∪ ciun 4455   class class class wbr 4583   × cxp 5036  Fun wfun 5798  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  1_𝑜c1o 7440   ≈ cen 7838   ↾_t crest 15904  Topctop 20517  TopOnctopon 20518  clsccl 20632  neicnei 20711  Hauscha 20922  Filcfil 21459   fLimf cflf 21549  CnExtccnext 21673

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-1o 7447  df-map 7746  df-pm 7747  df-en 7842  df-rest 15906  df-fbas 19564  df-top 20521  df-topon 20523  df-cld 20633  df-ntr 20634  df-cls 20635  df-nei 20712  df-haus 20929  df-fil 21460  df-flim 21553  df-flf 21554  df-cnext 21674

This theorem is referenced by:  cnextcn  21681  cnextfres1  21682