Theorem cnextfvval 22371

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 473	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → 𝐽 ∈ Top)
3		cnextf.4	. . . 4 ⊢ (𝜑 → 𝐾 ∈ Haus)
4	3	adantr 473	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → 𝐾 ∈ Haus)
5		cnextf.5	. . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
6	5	adantr 473	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → 𝐹:𝐴⟶𝐵)
7		cnextf.a	. . . 4 ⊢ (𝜑 → 𝐴 ⊆ 𝐶)
8	7	adantr 473	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → 𝐴 ⊆ 𝐶)
9		cnextf.1	. . . 4 ⊢ 𝐶 = ∪ 𝐽
10		cnextf.2	. . . 4 ⊢ 𝐵 = ∪ 𝐾
11	9, 10	cnextfun 22370	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Haus) ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ 𝐶)) → Fun ((𝐽CnExt𝐾)‘𝐹))
12	2, 4, 6, 8, 11	syl22anc 826	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → Fun ((𝐽CnExt𝐾)‘𝐹))
13		cnextf.6	. . . . . 6 ⊢ (𝜑 → ((cls‘𝐽)‘𝐴) = 𝐶)
14	13	eleq2d 2845	. . . . 5 ⊢ (𝜑 → (𝑋 ∈ ((cls‘𝐽)‘𝐴) ↔ 𝑋 ∈ 𝐶))
15	14	biimpar 470	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → 𝑋 ∈ ((cls‘𝐽)‘𝐴))
16		fvex 6506	. . . . . . 7 ⊢ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹) ∈ V
17	16	uniex 7277	. . . . . 6 ⊢ ∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹) ∈ V
18	17	snid 4467	. . . . 5 ⊢ ∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹) ∈ {∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹)}
19		sneq 4445	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑋 → {𝑥} = {𝑋})
20	19	fveq2d 6497	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑋 → ((nei‘𝐽)‘{𝑥}) = ((nei‘𝐽)‘{𝑋}))
21	20	oveq1d 6985	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑋 → (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) = (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))
22	21	oveq2d 6986	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → (𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴)) = (𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴)))
23	22	fveq1d 6495	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) = ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹))
24	23	breq1d 4933	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) ≈ 1o ↔ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹) ≈ 1o))
25	24	imbi2d 333	. . . . . . . 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 473	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐾 ∈ Haus)
27	1	adantr 473	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐽 ∈ Top)
28	9	toptopon 21223	. . . . . . . . . . . 12 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝐶))
29	27, 28	sylib 210	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐽 ∈ (TopOn‘𝐶))
30	7	adantr 473	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐴 ⊆ 𝐶)
31		simpr 477	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝑥 ∈ 𝐶)
32	13	eleq2d 2845	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↔ 𝑥 ∈ 𝐶))
33	32	biimpar 470	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝑥 ∈ ((cls‘𝐽)‘𝐴))
34		trnei 22198	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ (TopOn‘𝐶) ∧ 𝐴 ⊆ 𝐶 ∧ 𝑥 ∈ 𝐶) → (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↔ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (Fil‘𝐴)))
35	34	biimpa 469	. . . . . . . . . . 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 1353	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (Fil‘𝐴))
37	5	adantr 473	. . . . . . . . . 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 22304	. . . . . . . . . 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 1353	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) ≈ 1o)
41	40	expcom 406	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐶 → (𝜑 → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) ≈ 1o))
42	25, 41	vtoclga 3487	. . . . . . 7 ⊢ (𝑋 ∈ 𝐶 → (𝜑 → ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹) ≈ 1o))
43	42	impcom 399	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹) ≈ 1o)
44		en1b 8368	. . . . . 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 210	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹) = {∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹)})
46	18, 45	syl5eleqr 2867	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹) ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹))
47		nfiu1 4817	. . . . . . . 8 ⊢ Ⅎ𝑥∪ 𝑥 ∈ ((cls‘𝐽)‘𝐴)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹))
48	47	nfel2 2942	. . . . . . 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 1873	. . . . . . 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 1866	. . . . . 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 4671	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → ⟨𝑥, ∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹)⟩ = ⟨𝑋, ∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹)⟩)
52	51	eleq1d 2844	. . . . . . 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 2847	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↔ 𝑋 ∈ ((cls‘𝐽)‘𝐴)))
54	23	eleq2d 2845	. . . . . . . 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 621	. . . . . . 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 338	. . . . . 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 5463	. . . . . 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 3479	. . . . 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 474	. . . 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 700	. . 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 4924	. . . 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 21637	. . . . . . . 8 ⊢ (𝐾 ∈ Haus → 𝐾 ∈ Top)
63	3, 62	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ Top)
64	63	adantr 473	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → 𝐾 ∈ Top)
65	9, 10	cnextfval 22368	. . . . . 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 826	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ((𝐽CnExt𝐾)‘𝐹) = ∪ 𝑥 ∈ ((cls‘𝐽)‘𝐴)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)))
67	66	eleq2d 2845	. . . 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 275	. . 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 249	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → 𝑋((𝐽CnExt𝐾)‘𝐹)∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹))
70		funbrfv 6540	. 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 65	1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (((𝐽CnExt𝐾)‘𝐹)‘𝑋) = ∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 387   ∧ w3a 1068   = wceq 1507   ∈ wcel 2050   ≠ wne 2961   ⊆ wss 3823  ∅c0 4172  {csn 4435  ⟨cop 4441  ∪ cuni 4706  ∪ ciun 4786   class class class wbr 4923   × cxp 5399  Fun wfun 6176  ⟶wf 6178  ‘cfv 6182  (class class class)co 6970  1_oc1o 7892   ≈ cen 8297   ↾_t crest 16544  Topctop 21199  TopOnctopon 21216  clsccl 21324  neicnei 21403  Hauscha 21614  Filcfil 22151   fLimf cflf 22241  CnExtccnext 22365

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744  ax-rep 5043  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-nel 3068  df-ral 3087  df-rex 3088  df-reu 3089  df-rab 3091  df-v 3411  df-sbc 3676  df-csb 3781  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4707  df-int 4744  df-iun 4788  df-iin 4789  df-br 4924  df-opab 4986  df-mpt 5003  df-id 5306  df-xp 5407  df-rel 5408  df-cnv 5409  df-co 5410  df-dm 5411  df-rn 5412  df-res 5413  df-ima 5414  df-suc 6029  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-f1 6187  df-fo 6188  df-f1o 6189  df-fv 6190  df-ov 6973  df-oprab 6974  df-mpo 6975  df-1st 7495  df-2nd 7496  df-1o 7899  df-map 8202  df-pm 8203  df-en 8301  df-rest 16546  df-fbas 20238  df-top 21200  df-topon 21217  df-cld 21325  df-ntr 21326  df-cls 21327  df-nei 21404  df-haus 21621  df-fil 22152  df-flim 22245  df-flf 22246  df-cnext 22366

This theorem is referenced by:  cnextcn  22373  cnextfres1  22374