Theorem cnextfvval 22676

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 483	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → 𝐽 ∈ Top)
3		cnextf.4	. . . 4 ⊢ (𝜑 → 𝐾 ∈ Haus)
4	3	adantr 483	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → 𝐾 ∈ Haus)
5		cnextf.5	. . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
6	5	adantr 483	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → 𝐹:𝐴⟶𝐵)
7		cnextf.a	. . . 4 ⊢ (𝜑 → 𝐴 ⊆ 𝐶)
8	7	adantr 483	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → 𝐴 ⊆ 𝐶)
9		cnextf.1	. . . 4 ⊢ 𝐶 = ∪ 𝐽
10		cnextf.2	. . . 4 ⊢ 𝐵 = ∪ 𝐾
11	9, 10	cnextfun 22675	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Haus) ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ 𝐶)) → Fun ((𝐽CnExt𝐾)‘𝐹))
12	2, 4, 6, 8, 11	syl22anc 836	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → Fun ((𝐽CnExt𝐾)‘𝐹))
13		cnextf.6	. . . . . 6 ⊢ (𝜑 → ((cls‘𝐽)‘𝐴) = 𝐶)
14	13	eleq2d 2901	. . . . 5 ⊢ (𝜑 → (𝑋 ∈ ((cls‘𝐽)‘𝐴) ↔ 𝑋 ∈ 𝐶))
15	14	biimpar 480	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → 𝑋 ∈ ((cls‘𝐽)‘𝐴))
16		fvex 6686	. . . . . . 7 ⊢ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹) ∈ V
17	16	uniex 7470	. . . . . 6 ⊢ ∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹) ∈ V
18	17	snid 4604	. . . . 5 ⊢ ∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹) ∈ {∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹)}
19		sneq 4580	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑋 → {𝑥} = {𝑋})
20	19	fveq2d 6677	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑋 → ((nei‘𝐽)‘{𝑥}) = ((nei‘𝐽)‘{𝑋}))
21	20	oveq1d 7174	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑋 → (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) = (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))
22	21	oveq2d 7175	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → (𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴)) = (𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴)))
23	22	fveq1d 6675	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) = ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹))
24	23	breq1d 5079	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) ≈ 1o ↔ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹) ≈ 1o))
25	24	imbi2d 343	. . . . . . . 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 483	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐾 ∈ Haus)
27	1	adantr 483	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐽 ∈ Top)
28	9	toptopon 21528	. . . . . . . . . . . 12 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝐶))
29	27, 28	sylib 220	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐽 ∈ (TopOn‘𝐶))
30	7	adantr 483	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐴 ⊆ 𝐶)
31		simpr 487	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝑥 ∈ 𝐶)
32	13	eleq2d 2901	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↔ 𝑥 ∈ 𝐶))
33	32	biimpar 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝑥 ∈ ((cls‘𝐽)‘𝐴))
34		trnei 22503	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ (TopOn‘𝐶) ∧ 𝐴 ⊆ 𝐶 ∧ 𝑥 ∈ 𝐶) → (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↔ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (Fil‘𝐴)))
35	34	biimpa 479	. . . . . . . . . . 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 1369	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (Fil‘𝐴))
37	5	adantr 483	. . . . . . . . . 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 22609	. . . . . . . . . 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 1369	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) ≈ 1o)
41	40	expcom 416	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐶 → (𝜑 → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) ≈ 1o))
42	25, 41	vtoclga 3577	. . . . . . 7 ⊢ (𝑋 ∈ 𝐶 → (𝜑 → ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹) ≈ 1o))
43	42	impcom 410	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹) ≈ 1o)
44		en1b 8580	. . . . . 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 220	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹) = {∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹)})
46	18, 45	eleqtrrid 2923	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹) ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹))
47		nfiu1 4956	. . . . . . . 8 ⊢ Ⅎ𝑥∪ 𝑥 ∈ ((cls‘𝐽)‘𝐴)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹))
48	47	nfel2 2999	. . . . . . 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 1914	. . . . . . 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 1903	. . . . . 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 4806	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → ⟨𝑥, ∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹)⟩ = ⟨𝑋, ∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹)⟩)
52	51	eleq1d 2900	. . . . . . 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 2903	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↔ 𝑋 ∈ ((cls‘𝐽)‘𝐴)))
54	23	eleq2d 2901	. . . . . . . 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 632	. . . . . . 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 348	. . . . . 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 5622	. . . . . 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 3569	. . . . 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 484	. . . 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 711	. . 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 5070	. . . 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 21942	. . . . . . . 8 ⊢ (𝐾 ∈ Haus → 𝐾 ∈ Top)
63	3, 62	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ Top)
64	63	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → 𝐾 ∈ Top)
65	9, 10	cnextfval 22673	. . . . . 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 836	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ((𝐽CnExt𝐾)‘𝐹) = ∪ 𝑥 ∈ ((cls‘𝐽)‘𝐴)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)))
67	66	eleq2d 2901	. . . 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 285	. . 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 259	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → 𝑋((𝐽CnExt𝐾)‘𝐹)∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹))
70		funbrfv 6719	. 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 208   ∧ wa 398   ∧ w3a 1083   = wceq 1536   ∈ wcel 2113   ≠ wne 3019   ⊆ wss 3939  ∅c0 4294  {csn 4570  ⟨cop 4576  ∪ cuni 4841  ∪ ciun 4922   class class class wbr 5069   × cxp 5556  Fun wfun 6352  ⟶wf 6354  ‘cfv 6358  (class class class)co 7159  1_oc1o 8098   ≈ cen 8509   ↾_t crest 16697  Topctop 21504  TopOnctopon 21521  clsccl 21629  neicnei 21708  Hauscha 21919  Filcfil 22456   fLimf cflf 22546  CnExtccnext 22670

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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-rep 5193  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-nel 3127  df-ral 3146  df-rex 3147  df-reu 3148  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-int 4880  df-iun 4924  df-iin 4925  df-br 5070  df-opab 5132  df-mpt 5150  df-id 5463  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-suc 6200  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-ov 7162  df-oprab 7163  df-mpo 7164  df-1st 7692  df-2nd 7693  df-1o 8105  df-map 8411  df-pm 8412  df-en 8513  df-rest 16699  df-fbas 20545  df-top 21505  df-topon 21522  df-cld 21630  df-ntr 21631  df-cls 21632  df-nei 21709  df-haus 21926  df-fil 22457  df-flim 22550  df-flf 22551  df-cnext 22671

This theorem is referenced by:  cnextcn  22678  cnextfres1  22679