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Theorem cnextcn 23961

Description: Extension by continuity. Theorem 1 of [BourbakiTop1] p. I.57. Given a topology 𝐽 on 𝐶, a subset 𝐴 dense in 𝐶, this states a condition for 𝐹 from 𝐴 to a regular space 𝐾 to be extensible by continuity. (Contributed by Thierry Arnoux, 1-Jan-2018.)

**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) ≠ ∅)
cnextcn.8	⊢ (𝜑 → 𝐾 ∈ Reg)

**Assertion**
Ref	Expression
cnextcn	⊢ (𝜑 → ((𝐽CnExt𝐾)‘𝐹) ∈ (𝐽 Cn 𝐾))

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

Proof of Theorem cnextcn Dummy variables 𝑦 𝑏 𝑑 𝑢 𝑣 𝑧 𝑤 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpll 766	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) → 𝜑)
2		simpll 766	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ (𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)}) ∧ 𝑑 ∈ ((nei‘𝐽)‘{𝑥}) ∧ ((cls‘𝐾)‘(𝐹 “ (𝑑 ∩ 𝐴))) ⊆ 𝑤)) → 𝜑)
3		simpr3 1197	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ (𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)}) ∧ 𝑑 ∈ ((nei‘𝐽)‘{𝑥}) ∧ ((cls‘𝐾)‘(𝐹 “ (𝑑 ∩ 𝐴))) ⊆ 𝑤)) → ((cls‘𝐾)‘(𝐹 “ (𝑑 ∩ 𝐴))) ⊆ 𝑤)
4		cnextf.3	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐽 ∈ Top)
5	4	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ (𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)}) ∧ 𝑑 ∈ ((nei‘𝐽)‘{𝑥}) ∧ ((cls‘𝐾)‘(𝐹 “ (𝑑 ∩ 𝐴))) ⊆ 𝑤)) → 𝐽 ∈ Top)
6		simpr2 1196	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ (𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)}) ∧ 𝑑 ∈ ((nei‘𝐽)‘{𝑥}) ∧ ((cls‘𝐾)‘(𝐹 “ (𝑑 ∩ 𝐴))) ⊆ 𝑤)) → 𝑑 ∈ ((nei‘𝐽)‘{𝑥}))
7		neii2 23002	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝑑 ∈ ((nei‘𝐽)‘{𝑥})) → ∃𝑣 ∈ 𝐽 ({𝑥} ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑑))
8	5, 6, 7	syl2anc 584	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ (𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)}) ∧ 𝑑 ∈ ((nei‘𝐽)‘{𝑥}) ∧ ((cls‘𝐾)‘(𝐹 “ (𝑑 ∩ 𝐴))) ⊆ 𝑤)) → ∃𝑣 ∈ 𝐽 ({𝑥} ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑑))
9		vex 3454	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑥 ∈ V
10	9	snss 4752	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ 𝑣 ↔ {𝑥} ⊆ 𝑣)
11	10	biimpri 228	. . . . . . . . . . . . . . . . . 18 ⊢ ({𝑥} ⊆ 𝑣 → 𝑥 ∈ 𝑣)
12	11	anim1i 615	. . . . . . . . . . . . . . . . 17 ⊢ (({𝑥} ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑑) → (𝑥 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑑))
13	12	anim2i 617	. . . . . . . . . . . . . . . 16 ⊢ ((𝑣 ∈ 𝐽 ∧ ({𝑥} ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑑)) → (𝑣 ∈ 𝐽 ∧ (𝑥 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑑)))
14	13	anim2i 617	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐽 ∧ ({𝑥} ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑑))) → (𝜑 ∧ (𝑣 ∈ 𝐽 ∧ (𝑥 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑑))))
15	14	ex 412	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑣 ∈ 𝐽 ∧ ({𝑥} ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑑)) → (𝜑 ∧ (𝑣 ∈ 𝐽 ∧ (𝑥 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑑)))))
16		3anass 1094	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐽 ∧ 𝑥 ∈ 𝑣) ↔ (𝜑 ∧ (𝑣 ∈ 𝐽 ∧ 𝑥 ∈ 𝑣)))
17	16	anbi1i 624	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽 ∧ 𝑥 ∈ 𝑣) ∧ 𝑣 ⊆ 𝑑) ↔ ((𝜑 ∧ (𝑣 ∈ 𝐽 ∧ 𝑥 ∈ 𝑣)) ∧ 𝑣 ⊆ 𝑑))
18		anass 468	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐽 ∧ 𝑥 ∈ 𝑣)) ∧ 𝑣 ⊆ 𝑑) ↔ (𝜑 ∧ ((𝑣 ∈ 𝐽 ∧ 𝑥 ∈ 𝑣) ∧ 𝑣 ⊆ 𝑑)))
19		anass 468	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑣 ∈ 𝐽 ∧ 𝑥 ∈ 𝑣) ∧ 𝑣 ⊆ 𝑑) ↔ (𝑣 ∈ 𝐽 ∧ (𝑥 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑑)))
20	19	anbi2i 623	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑣 ∈ 𝐽 ∧ 𝑥 ∈ 𝑣) ∧ 𝑣 ⊆ 𝑑)) ↔ (𝜑 ∧ (𝑣 ∈ 𝐽 ∧ (𝑥 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑑))))
21	17, 18, 20	3bitri 297	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽 ∧ 𝑥 ∈ 𝑣) ∧ 𝑣 ⊆ 𝑑) ↔ (𝜑 ∧ (𝑣 ∈ 𝐽 ∧ (𝑥 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑑))))
22		opnneip 23013	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐽 ∈ Top ∧ 𝑣 ∈ 𝐽 ∧ 𝑥 ∈ 𝑣) → 𝑣 ∈ ((nei‘𝐽)‘{𝑥}))
23	4, 22	syl3an1 1163	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐽 ∧ 𝑥 ∈ 𝑣) → 𝑣 ∈ ((nei‘𝐽)‘{𝑥}))
24	23	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽 ∧ 𝑥 ∈ 𝑣) ∧ 𝑣 ⊆ 𝑑) → 𝑣 ∈ ((nei‘𝐽)‘{𝑥}))
25		simpr2 1196	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑣 ⊆ 𝑑 ∧ (𝜑 ∧ 𝑣 ∈ 𝐽 ∧ 𝑥 ∈ 𝑣)) → 𝑣 ∈ 𝐽)
26	25	ex 412	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 ⊆ 𝑑 → ((𝜑 ∧ 𝑣 ∈ 𝐽 ∧ 𝑥 ∈ 𝑣) → 𝑣 ∈ 𝐽))
27	26	imdistanri 569	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽 ∧ 𝑥 ∈ 𝑣) ∧ 𝑣 ⊆ 𝑑) → (𝑣 ∈ 𝐽 ∧ 𝑣 ⊆ 𝑑))
28	24, 27	jca 511	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽 ∧ 𝑥 ∈ 𝑣) ∧ 𝑣 ⊆ 𝑑) → (𝑣 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝑣 ∈ 𝐽 ∧ 𝑣 ⊆ 𝑑)))
29	21, 28	sylbir 235	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐽 ∧ (𝑥 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑑))) → (𝑣 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝑣 ∈ 𝐽 ∧ 𝑣 ⊆ 𝑑)))
30	15, 29	syl6 35	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑣 ∈ 𝐽 ∧ ({𝑥} ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑑)) → (𝑣 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝑣 ∈ 𝐽 ∧ 𝑣 ⊆ 𝑑))))
31	30	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((cls‘𝐾)‘(𝐹 “ (𝑑 ∩ 𝐴))) ⊆ 𝑤) → ((𝑣 ∈ 𝐽 ∧ ({𝑥} ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑑)) → (𝑣 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝑣 ∈ 𝐽 ∧ 𝑣 ⊆ 𝑑))))
32		cnextf.4	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐾 ∈ Haus)
33		haustop 23225	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐾 ∈ Haus → 𝐾 ∈ Top)
34	32, 33	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐾 ∈ Top)
35		imassrn 6045	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐹 “ (𝑑 ∩ 𝐴)) ⊆ ran 𝐹
36		cnextf.5	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
37	36	frnd 6699	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ran 𝐹 ⊆ 𝐵)
38	35, 37	sstrid 3961	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐹 “ (𝑑 ∩ 𝐴)) ⊆ 𝐵)
39		ssrin 4208	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑣 ⊆ 𝑑 → (𝑣 ∩ 𝐴) ⊆ (𝑑 ∩ 𝐴))
40		imass2 6076	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑣 ∩ 𝐴) ⊆ (𝑑 ∩ 𝐴) → (𝐹 “ (𝑣 ∩ 𝐴)) ⊆ (𝐹 “ (𝑑 ∩ 𝐴)))
41	39, 40	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 ⊆ 𝑑 → (𝐹 “ (𝑣 ∩ 𝐴)) ⊆ (𝐹 “ (𝑑 ∩ 𝐴)))
42		cnextf.2	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐵 = ∪ 𝐾
43	42	clsss 22948	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐾 ∈ Top ∧ (𝐹 “ (𝑑 ∩ 𝐴)) ⊆ 𝐵 ∧ (𝐹 “ (𝑣 ∩ 𝐴)) ⊆ (𝐹 “ (𝑑 ∩ 𝐴))) → ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))) ⊆ ((cls‘𝐾)‘(𝐹 “ (𝑑 ∩ 𝐴))))
44	34, 38, 41, 43	syl2an3an 1424	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑣 ⊆ 𝑑) → ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))) ⊆ ((cls‘𝐾)‘(𝐹 “ (𝑑 ∩ 𝐴))))
45		sstr 3958	. . . . . . . . . . . . . . . . 17 ⊢ ((((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))) ⊆ ((cls‘𝐾)‘(𝐹 “ (𝑑 ∩ 𝐴))) ∧ ((cls‘𝐾)‘(𝐹 “ (𝑑 ∩ 𝐴))) ⊆ 𝑤) → ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))) ⊆ 𝑤)
46	44, 45	sylan 580	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑣 ⊆ 𝑑) ∧ ((cls‘𝐾)‘(𝐹 “ (𝑑 ∩ 𝐴))) ⊆ 𝑤) → ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))) ⊆ 𝑤)
47	46	an32s 652	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ((cls‘𝐾)‘(𝐹 “ (𝑑 ∩ 𝐴))) ⊆ 𝑤) ∧ 𝑣 ⊆ 𝑑) → ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))) ⊆ 𝑤)
48	47	ex 412	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((cls‘𝐾)‘(𝐹 “ (𝑑 ∩ 𝐴))) ⊆ 𝑤) → (𝑣 ⊆ 𝑑 → ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))) ⊆ 𝑤))
49	48	anim2d 612	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((cls‘𝐾)‘(𝐹 “ (𝑑 ∩ 𝐴))) ⊆ 𝑤) → ((𝑣 ∈ 𝐽 ∧ 𝑣 ⊆ 𝑑) → (𝑣 ∈ 𝐽 ∧ ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))) ⊆ 𝑤)))
50	49	anim2d 612	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((cls‘𝐾)‘(𝐹 “ (𝑑 ∩ 𝐴))) ⊆ 𝑤) → ((𝑣 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝑣 ∈ 𝐽 ∧ 𝑣 ⊆ 𝑑)) → (𝑣 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝑣 ∈ 𝐽 ∧ ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))) ⊆ 𝑤))))
51	31, 50	syld 47	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((cls‘𝐾)‘(𝐹 “ (𝑑 ∩ 𝐴))) ⊆ 𝑤) → ((𝑣 ∈ 𝐽 ∧ ({𝑥} ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑑)) → (𝑣 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝑣 ∈ 𝐽 ∧ ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))) ⊆ 𝑤))))
52	51	reximdv2 3144	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((cls‘𝐾)‘(𝐹 “ (𝑑 ∩ 𝐴))) ⊆ 𝑤) → (∃𝑣 ∈ 𝐽 ({𝑥} ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑑) → ∃𝑣 ∈ ((nei‘𝐽)‘{𝑥})(𝑣 ∈ 𝐽 ∧ ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))) ⊆ 𝑤)))
53	52	imp 406	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((cls‘𝐾)‘(𝐹 “ (𝑑 ∩ 𝐴))) ⊆ 𝑤) ∧ ∃𝑣 ∈ 𝐽 ({𝑥} ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑑)) → ∃𝑣 ∈ ((nei‘𝐽)‘{𝑥})(𝑣 ∈ 𝐽 ∧ ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))) ⊆ 𝑤))
54	2, 3, 8, 53	syl21anc 837	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ (𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)}) ∧ 𝑑 ∈ ((nei‘𝐽)‘{𝑥}) ∧ ((cls‘𝐾)‘(𝐹 “ (𝑑 ∩ 𝐴))) ⊆ 𝑤)) → ∃𝑣 ∈ ((nei‘𝐽)‘{𝑥})(𝑣 ∈ 𝐽 ∧ ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))) ⊆ 𝑤))
55	54	3anassrs 1361	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) ∧ 𝑑 ∈ ((nei‘𝐽)‘{𝑥})) ∧ ((cls‘𝐾)‘(𝐹 “ (𝑑 ∩ 𝐴))) ⊆ 𝑤) → ∃𝑣 ∈ ((nei‘𝐽)‘{𝑥})(𝑣 ∈ 𝐽 ∧ ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))) ⊆ 𝑤))
56		simpr 484	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) ∧ 𝑢 ∈ (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴)) ∧ ((cls‘𝐾)‘(𝐹 “ 𝑢)) ⊆ 𝑤) → ((cls‘𝐾)‘(𝐹 “ 𝑢)) ⊆ 𝑤)
57		simp-4l 782	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) ∧ 𝑢 ∈ (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴)) ∧ ((cls‘𝐾)‘(𝐹 “ 𝑢)) ⊆ 𝑤) → 𝜑)
58		simplr 768	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) ∧ 𝑢 ∈ (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴)) ∧ ((cls‘𝐾)‘(𝐹 “ 𝑢)) ⊆ 𝑤) → 𝑢 ∈ (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴))
59		imaeq2 6030	. . . . . . . . . . . . . 14 ⊢ (𝑢 = (𝑑 ∩ 𝐴) → (𝐹 “ 𝑢) = (𝐹 “ (𝑑 ∩ 𝐴)))
60	59	fveq2d 6865	. . . . . . . . . . . . 13 ⊢ (𝑢 = (𝑑 ∩ 𝐴) → ((cls‘𝐾)‘(𝐹 “ 𝑢)) = ((cls‘𝐾)‘(𝐹 “ (𝑑 ∩ 𝐴))))
61	60	sseq1d 3981	. . . . . . . . . . . 12 ⊢ (𝑢 = (𝑑 ∩ 𝐴) → (((cls‘𝐾)‘(𝐹 “ 𝑢)) ⊆ 𝑤 ↔ ((cls‘𝐾)‘(𝐹 “ (𝑑 ∩ 𝐴))) ⊆ 𝑤))
62	61	biimpcd 249	. . . . . . . . . . 11 ⊢ (((cls‘𝐾)‘(𝐹 “ 𝑢)) ⊆ 𝑤 → (𝑢 = (𝑑 ∩ 𝐴) → ((cls‘𝐾)‘(𝐹 “ (𝑑 ∩ 𝐴))) ⊆ 𝑤))
63	62	reximdv 3149	. . . . . . . . . 10 ⊢ (((cls‘𝐾)‘(𝐹 “ 𝑢)) ⊆ 𝑤 → (∃𝑑 ∈ ((nei‘𝐽)‘{𝑥})𝑢 = (𝑑 ∩ 𝐴) → ∃𝑑 ∈ ((nei‘𝐽)‘{𝑥})((cls‘𝐾)‘(𝐹 “ (𝑑 ∩ 𝐴))) ⊆ 𝑤))
64		fvexd 6876	. . . . . . . . . . . 12 ⊢ (𝜑 → ((nei‘𝐽)‘{𝑥}) ∈ V)
65		cnextf.1	. . . . . . . . . . . . . . . 16 ⊢ 𝐶 = ∪ 𝐽
66	65	toptopon 22811	. . . . . . . . . . . . . . 15 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝐶))
67	4, 66	sylib 218	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝐶))
68	67	elfvexd 6900	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐶 ∈ V)
69		cnextf.a	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 ⊆ 𝐶)
70	68, 69	ssexd 5282	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ V)
71		elrest 17397	. . . . . . . . . . . 12 ⊢ ((((nei‘𝐽)‘{𝑥}) ∈ V ∧ 𝐴 ∈ V) → (𝑢 ∈ (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴) ↔ ∃𝑑 ∈ ((nei‘𝐽)‘{𝑥})𝑢 = (𝑑 ∩ 𝐴)))
72	64, 70, 71	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑢 ∈ (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴) ↔ ∃𝑑 ∈ ((nei‘𝐽)‘{𝑥})𝑢 = (𝑑 ∩ 𝐴)))
73	72	biimpa 476	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑢 ∈ (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴)) → ∃𝑑 ∈ ((nei‘𝐽)‘{𝑥})𝑢 = (𝑑 ∩ 𝐴))
74	63, 73	impel 505	. . . . . . . . 9 ⊢ ((((cls‘𝐾)‘(𝐹 “ 𝑢)) ⊆ 𝑤 ∧ (𝜑 ∧ 𝑢 ∈ (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴))) → ∃𝑑 ∈ ((nei‘𝐽)‘{𝑥})((cls‘𝐾)‘(𝐹 “ (𝑑 ∩ 𝐴))) ⊆ 𝑤)
75	56, 57, 58, 74	syl12anc 836	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) ∧ 𝑢 ∈ (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴)) ∧ ((cls‘𝐾)‘(𝐹 “ 𝑢)) ⊆ 𝑤) → ∃𝑑 ∈ ((nei‘𝐽)‘{𝑥})((cls‘𝐾)‘(𝐹 “ (𝑑 ∩ 𝐴))) ⊆ 𝑤)
76		cnextf.6	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((cls‘𝐽)‘𝐴) = 𝐶)
77		eleq1w 2812	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐶 ↔ 𝑦 ∈ 𝐶))
78	77	anbi2d 630	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑦 → ((𝜑 ∧ 𝑥 ∈ 𝐶) ↔ (𝜑 ∧ 𝑦 ∈ 𝐶)))
79		sneq 4602	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦})
80	79	fveq2d 6865	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑦 → ((nei‘𝐽)‘{𝑥}) = ((nei‘𝐽)‘{𝑦}))
81	80	oveq1d 7405	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = 𝑦 → (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴) = (((nei‘𝐽)‘{𝑦}) ↾_t 𝐴))
82	81	oveq2d 7406	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑦 → (𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴)) = (𝐾 fLimf (((nei‘𝐽)‘{𝑦}) ↾_t 𝐴)))
83	82	fveq1d 6863	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑦 → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴))‘𝐹) = ((𝐾 fLimf (((nei‘𝐽)‘{𝑦}) ↾_t 𝐴))‘𝐹))
84	83	neeq1d 2985	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑦 → (((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴))‘𝐹) ≠ ∅ ↔ ((𝐾 fLimf (((nei‘𝐽)‘{𝑦}) ↾_t 𝐴))‘𝐹) ≠ ∅))
85	78, 84	imbi12d 344	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → (((𝜑 ∧ 𝑥 ∈ 𝐶) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴))‘𝐹) ≠ ∅) ↔ ((𝜑 ∧ 𝑦 ∈ 𝐶) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑦}) ↾_t 𝐴))‘𝐹) ≠ ∅)))
86		cnextf.7	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴))‘𝐹) ≠ ∅)
87	85, 86	chvarvv 1989	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑦}) ↾_t 𝐴))‘𝐹) ≠ ∅)
88	65, 42, 4, 32, 36, 69, 76, 87	cnextfvval 23959	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (((𝐽CnExt𝐾)‘𝐹)‘𝑥) = ∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴))‘𝐹))
89		fvex 6874	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴))‘𝐹) ∈ V
90	89	uniex 7720	. . . . . . . . . . . . . . . . 17 ⊢ ∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴))‘𝐹) ∈ V
91	90	snid 4629	. . . . . . . . . . . . . . . 16 ⊢ ∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴))‘𝐹) ∈ {∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴))‘𝐹)}
92	32	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐾 ∈ Haus)
93	76	eleq2d 2815	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↔ 𝑥 ∈ 𝐶))
94	93	biimpar 477	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝑥 ∈ ((cls‘𝐽)‘𝐴))
95	67	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐽 ∈ (TopOn‘𝐶))
96	69	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐴 ⊆ 𝐶)
97		simpr 484	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝑥 ∈ 𝐶)
98		trnei 23786	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐽 ∈ (TopOn‘𝐶) ∧ 𝐴 ⊆ 𝐶 ∧ 𝑥 ∈ 𝐶) → (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↔ (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴) ∈ (Fil‘𝐴)))
99	95, 96, 97, 98	syl3anc 1373	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↔ (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴) ∈ (Fil‘𝐴)))
100	94, 99	mpbid 232	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴) ∈ (Fil‘𝐴))
101	36	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐹:𝐴⟶𝐵)
102	42	hausflf2 23892	. . . . . . . . . . . . . . . . . 18 ⊢ (((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) ≈ 1_o)
103	92, 100, 101, 86, 102	syl31anc 1375	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴))‘𝐹) ≈ 1_o)
104		en1b 8999	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴))‘𝐹) ≈ 1_o ↔ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴))‘𝐹) = {∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴))‘𝐹)})
105	103, 104	sylib 218	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴))‘𝐹) = {∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴))‘𝐹)})
106	91, 105	eleqtrrid 2836	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴))‘𝐹) ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴))‘𝐹))
107	88, 106	eqeltrd 2829	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴))‘𝐹))
108	42	toptopon 22811	. . . . . . . . . . . . . . . . 17 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝐵))
109	34, 108	sylib 218	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝐵))
110	109	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐾 ∈ (TopOn‘𝐵))
111		flfnei 23885	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ (TopOn‘𝐵) ∧ (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴) ∈ (Fil‘𝐴) ∧ 𝐹:𝐴⟶𝐵) → ((((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴))‘𝐹) ↔ ((((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝐵 ∧ ∀𝑏 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})∃𝑢 ∈ (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴)(𝐹 “ 𝑢) ⊆ 𝑏)))
112	110, 100, 101, 111	syl3anc 1373	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴))‘𝐹) ↔ ((((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝐵 ∧ ∀𝑏 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})∃𝑢 ∈ (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴)(𝐹 “ 𝑢) ⊆ 𝑏)))
113	107, 112	mpbid 232	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝐵 ∧ ∀𝑏 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})∃𝑢 ∈ (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴)(𝐹 “ 𝑢) ⊆ 𝑏))
114	113	simprd 495	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ∀𝑏 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})∃𝑢 ∈ (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴)(𝐹 “ 𝑢) ⊆ 𝑏)
115	114	r19.21bi 3230	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑏 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) → ∃𝑢 ∈ (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴)(𝐹 “ 𝑢) ⊆ 𝑏)
116	115	ad4ant13 751	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) ∧ 𝑏 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑤) → ∃𝑢 ∈ (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴)(𝐹 “ 𝑢) ⊆ 𝑏)
117	34	ad3antrrr 730	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑏 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑤) → 𝐾 ∈ Top)
118		simplr 768	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑏 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑤) → 𝑏 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)}))
119	42	neii1 23000	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ Top ∧ 𝑏 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) → 𝑏 ⊆ 𝐵)
120	117, 118, 119	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑏 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑤) → 𝑏 ⊆ 𝐵)
121		simpr 484	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑏 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑤) → ((cls‘𝐾)‘𝑏) ⊆ 𝑤)
122	42	clsss 22948	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ Top ∧ 𝑏 ⊆ 𝐵 ∧ (𝐹 “ 𝑢) ⊆ 𝑏) → ((cls‘𝐾)‘(𝐹 “ 𝑢)) ⊆ ((cls‘𝐾)‘𝑏))
123		sstr 3958	. . . . . . . . . . . . . . . 16 ⊢ ((((cls‘𝐾)‘(𝐹 “ 𝑢)) ⊆ ((cls‘𝐾)‘𝑏) ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑤) → ((cls‘𝐾)‘(𝐹 “ 𝑢)) ⊆ 𝑤)
124	122, 123	sylan 580	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ Top ∧ 𝑏 ⊆ 𝐵 ∧ (𝐹 “ 𝑢) ⊆ 𝑏) ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑤) → ((cls‘𝐾)‘(𝐹 “ 𝑢)) ⊆ 𝑤)
125	124	3an1rs 1360	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ Top ∧ 𝑏 ⊆ 𝐵 ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑤) ∧ (𝐹 “ 𝑢) ⊆ 𝑏) → ((cls‘𝐾)‘(𝐹 “ 𝑢)) ⊆ 𝑤)
126	125	ex 412	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ Top ∧ 𝑏 ⊆ 𝐵 ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑤) → ((𝐹 “ 𝑢) ⊆ 𝑏 → ((cls‘𝐾)‘(𝐹 “ 𝑢)) ⊆ 𝑤))
127	126	reximdv 3149	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ Top ∧ 𝑏 ⊆ 𝐵 ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑤) → (∃𝑢 ∈ (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴)(𝐹 “ 𝑢) ⊆ 𝑏 → ∃𝑢 ∈ (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴)((cls‘𝐾)‘(𝐹 “ 𝑢)) ⊆ 𝑤))
128	117, 120, 121, 127	syl3anc 1373	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑏 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑤) → (∃𝑢 ∈ (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴)(𝐹 “ 𝑢) ⊆ 𝑏 → ∃𝑢 ∈ (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴)((cls‘𝐾)‘(𝐹 “ 𝑢)) ⊆ 𝑤))
129	128	adantllr 719	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) ∧ 𝑏 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑤) → (∃𝑢 ∈ (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴)(𝐹 “ 𝑢) ⊆ 𝑏 → ∃𝑢 ∈ (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴)((cls‘𝐾)‘(𝐹 “ 𝑢)) ⊆ 𝑤))
130	116, 129	mpd 15	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) ∧ 𝑏 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑤) → ∃𝑢 ∈ (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴)((cls‘𝐾)‘(𝐹 “ 𝑢)) ⊆ 𝑤)
131	34	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) → 𝐾 ∈ Top)
132		cnextcn.8	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐾 ∈ Reg)
133	132	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) → 𝐾 ∈ Reg)
134	133	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) ∧ 𝑐 ∈ 𝐾) ∧ ((((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑐 ∧ 𝑐 ⊆ 𝑤)) → 𝐾 ∈ Reg)
135		simplr 768	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) ∧ 𝑐 ∈ 𝐾) ∧ ((((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑐 ∧ 𝑐 ⊆ 𝑤)) → 𝑐 ∈ 𝐾)
136		simprl 770	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) ∧ 𝑐 ∈ 𝐾) ∧ ((((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑐 ∧ 𝑐 ⊆ 𝑤)) → (((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑐)
137		regsep 23228	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ Reg ∧ 𝑐 ∈ 𝐾 ∧ (((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑐) → ∃𝑏 ∈ 𝐾 ((((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑏 ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑐))
138	134, 135, 136, 137	syl3anc 1373	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) ∧ 𝑐 ∈ 𝐾) ∧ ((((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑐 ∧ 𝑐 ⊆ 𝑤)) → ∃𝑏 ∈ 𝐾 ((((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑏 ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑐))
139		sstr 3958	. . . . . . . . . . . . . . . 16 ⊢ ((((cls‘𝐾)‘𝑏) ⊆ 𝑐 ∧ 𝑐 ⊆ 𝑤) → ((cls‘𝐾)‘𝑏) ⊆ 𝑤)
140	139	expcom 413	. . . . . . . . . . . . . . 15 ⊢ (𝑐 ⊆ 𝑤 → (((cls‘𝐾)‘𝑏) ⊆ 𝑐 → ((cls‘𝐾)‘𝑏) ⊆ 𝑤))
141	140	anim2d 612	. . . . . . . . . . . . . 14 ⊢ (𝑐 ⊆ 𝑤 → (((((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑏 ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑐) → ((((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑏 ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑤)))
142	141	reximdv 3149	. . . . . . . . . . . . 13 ⊢ (𝑐 ⊆ 𝑤 → (∃𝑏 ∈ 𝐾 ((((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑏 ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑐) → ∃𝑏 ∈ 𝐾 ((((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑏 ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑤)))
143	142	ad2antll 729	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) ∧ 𝑐 ∈ 𝐾) ∧ ((((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑐 ∧ 𝑐 ⊆ 𝑤)) → (∃𝑏 ∈ 𝐾 ((((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑏 ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑐) → ∃𝑏 ∈ 𝐾 ((((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑏 ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑤)))
144	138, 143	mpd 15	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) ∧ 𝑐 ∈ 𝐾) ∧ ((((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑐 ∧ 𝑐 ⊆ 𝑤)) → ∃𝑏 ∈ 𝐾 ((((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑏 ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑤))
145		simpr 484	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) → 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)}))
146		neii2 23002	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ Top ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) → ∃𝑐 ∈ 𝐾 ({(((𝐽CnExt𝐾)‘𝐹)‘𝑥)} ⊆ 𝑐 ∧ 𝑐 ⊆ 𝑤))
147		fvex 6874	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ V
148	147	snss 4752	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑐 ↔ {(((𝐽CnExt𝐾)‘𝐹)‘𝑥)} ⊆ 𝑐)
149	148	anbi1i 624	. . . . . . . . . . . . . . 15 ⊢ (((((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑐 ∧ 𝑐 ⊆ 𝑤) ↔ ({(((𝐽CnExt𝐾)‘𝐹)‘𝑥)} ⊆ 𝑐 ∧ 𝑐 ⊆ 𝑤))
150	149	biimpri 228	. . . . . . . . . . . . . 14 ⊢ (({(((𝐽CnExt𝐾)‘𝐹)‘𝑥)} ⊆ 𝑐 ∧ 𝑐 ⊆ 𝑤) → ((((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑐 ∧ 𝑐 ⊆ 𝑤))
151	150	reximi 3068	. . . . . . . . . . . . 13 ⊢ (∃𝑐 ∈ 𝐾 ({(((𝐽CnExt𝐾)‘𝐹)‘𝑥)} ⊆ 𝑐 ∧ 𝑐 ⊆ 𝑤) → ∃𝑐 ∈ 𝐾 ((((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑐 ∧ 𝑐 ⊆ 𝑤))
152	146, 151	syl 17	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ Top ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) → ∃𝑐 ∈ 𝐾 ((((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑐 ∧ 𝑐 ⊆ 𝑤))
153	131, 145, 152	syl2anc 584	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) → ∃𝑐 ∈ 𝐾 ((((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑐 ∧ 𝑐 ⊆ 𝑤))
154	144, 153	r19.29a 3142	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) → ∃𝑏 ∈ 𝐾 ((((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑏 ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑤))
155		anass 468	. . . . . . . . . . . 12 ⊢ (((𝑏 ∈ 𝐾 ∧ (((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑏) ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑤) ↔ (𝑏 ∈ 𝐾 ∧ ((((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑏 ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑤)))
156		opnneip 23013	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ Top ∧ 𝑏 ∈ 𝐾 ∧ (((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑏) → 𝑏 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)}))
157	156	3expib 1122	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ Top → ((𝑏 ∈ 𝐾 ∧ (((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑏) → 𝑏 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})))
158	157	anim1d 611	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ Top → (((𝑏 ∈ 𝐾 ∧ (((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑏) ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑤) → (𝑏 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)}) ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑤)))
159	155, 158	biimtrrid 243	. . . . . . . . . . 11 ⊢ (𝐾 ∈ Top → ((𝑏 ∈ 𝐾 ∧ ((((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑏 ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑤)) → (𝑏 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)}) ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑤)))
160	159	reximdv2 3144	. . . . . . . . . 10 ⊢ (𝐾 ∈ Top → (∃𝑏 ∈ 𝐾 ((((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑏 ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑤) → ∃𝑏 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})((cls‘𝐾)‘𝑏) ⊆ 𝑤))
161	131, 154, 160	sylc 65	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) → ∃𝑏 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})((cls‘𝐾)‘𝑏) ⊆ 𝑤)
162	130, 161	r19.29a 3142	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) → ∃𝑢 ∈ (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴)((cls‘𝐾)‘(𝐹 “ 𝑢)) ⊆ 𝑤)
163	75, 162	r19.29a 3142	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) → ∃𝑑 ∈ ((nei‘𝐽)‘{𝑥})((cls‘𝐾)‘(𝐹 “ (𝑑 ∩ 𝐴))) ⊆ 𝑤)
164	55, 163	r19.29a 3142	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) → ∃𝑣 ∈ ((nei‘𝐽)‘{𝑥})(𝑣 ∈ 𝐽 ∧ ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))) ⊆ 𝑤))
165		simplr 768	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))) ⊆ 𝑤) ∧ 𝑧 ∈ 𝑣) → ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))) ⊆ 𝑤)
166		simpll 766	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ 𝑧 ∈ 𝑣) → 𝜑)
167	4	ad2antrr 726	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ 𝑧 ∈ 𝑣) → 𝐽 ∈ Top)
168		simplr 768	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ 𝑧 ∈ 𝑣) → 𝑣 ∈ 𝐽)
169	65	eltopss 22801	. . . . . . . . . . . . . . 15 ⊢ ((𝐽 ∈ Top ∧ 𝑣 ∈ 𝐽) → 𝑣 ⊆ 𝐶)
170	167, 168, 169	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ 𝑧 ∈ 𝑣) → 𝑣 ⊆ 𝐶)
171		simpr 484	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ 𝑧 ∈ 𝑣) → 𝑧 ∈ 𝑣)
172	170, 171	sseldd 3950	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ 𝑧 ∈ 𝑣) → 𝑧 ∈ 𝐶)
173		fvexd 6876	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ 𝑧 ∈ 𝑣) → ((nei‘𝐽)‘{𝑧}) ∈ V)
174	70	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ 𝑧 ∈ 𝑣) → 𝐴 ∈ V)
175		opnneip 23013	. . . . . . . . . . . . . . . 16 ⊢ ((𝐽 ∈ Top ∧ 𝑣 ∈ 𝐽 ∧ 𝑧 ∈ 𝑣) → 𝑣 ∈ ((nei‘𝐽)‘{𝑧}))
176	4, 175	syl3an1 1163	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐽 ∧ 𝑧 ∈ 𝑣) → 𝑣 ∈ ((nei‘𝐽)‘{𝑧}))
177	176	3expa 1118	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ 𝑧 ∈ 𝑣) → 𝑣 ∈ ((nei‘𝐽)‘{𝑧}))
178		elrestr 17398	. . . . . . . . . . . . . 14 ⊢ ((((nei‘𝐽)‘{𝑧}) ∈ V ∧ 𝐴 ∈ V ∧ 𝑣 ∈ ((nei‘𝐽)‘{𝑧})) → (𝑣 ∩ 𝐴) ∈ (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴))
179	173, 174, 177, 178	syl3anc 1373	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ 𝑧 ∈ 𝑣) → (𝑣 ∩ 𝐴) ∈ (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴))
180	65, 42, 4, 32, 36, 69, 76, 86	cnextfvval 23959	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → (((𝐽CnExt𝐾)‘𝐹)‘𝑧) = ∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴))‘𝐹))
181	180	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐶) ∧ (𝑣 ∩ 𝐴) ∈ (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴)) → (((𝐽CnExt𝐾)‘𝐹)‘𝑧) = ∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴))‘𝐹))
182	32	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → 𝐾 ∈ Haus)
183	76	eleq2d 2815	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝑧 ∈ ((cls‘𝐽)‘𝐴) ↔ 𝑧 ∈ 𝐶))
184	183	biimpar 477	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → 𝑧 ∈ ((cls‘𝐽)‘𝐴))
185	67	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → 𝐽 ∈ (TopOn‘𝐶))
186	69	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → 𝐴 ⊆ 𝐶)
187		simpr 484	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → 𝑧 ∈ 𝐶)
188		trnei 23786	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐽 ∈ (TopOn‘𝐶) ∧ 𝐴 ⊆ 𝐶 ∧ 𝑧 ∈ 𝐶) → (𝑧 ∈ ((cls‘𝐽)‘𝐴) ↔ (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴) ∈ (Fil‘𝐴)))
189	185, 186, 187, 188	syl3anc 1373	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → (𝑧 ∈ ((cls‘𝐽)‘𝐴) ↔ (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴) ∈ (Fil‘𝐴)))
190	184, 189	mpbid 232	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴) ∈ (Fil‘𝐴))
191	36	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → 𝐹:𝐴⟶𝐵)
192		eleq1w 2812	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐶 ↔ 𝑧 ∈ 𝐶))
193	192	anbi2d 630	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = 𝑧 → ((𝜑 ∧ 𝑥 ∈ 𝐶) ↔ (𝜑 ∧ 𝑧 ∈ 𝐶)))
194		sneq 4602	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑥 = 𝑧 → {𝑥} = {𝑧})
195	194	fveq2d 6865	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥 = 𝑧 → ((nei‘𝐽)‘{𝑥}) = ((nei‘𝐽)‘{𝑧}))
196	195	oveq1d 7405	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 = 𝑧 → (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴) = (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴))
197	196	oveq2d 7406	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = 𝑧 → (𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴)) = (𝐾 fLimf (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴)))
198	197	fveq1d 6863	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑧 → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴))‘𝐹) = ((𝐾 fLimf (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴))‘𝐹))
199	198	neeq1d 2985	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = 𝑧 → (((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴))‘𝐹) ≠ ∅ ↔ ((𝐾 fLimf (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴))‘𝐹) ≠ ∅))
200	193, 199	imbi12d 344	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑧 → (((𝜑 ∧ 𝑥 ∈ 𝐶) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴))‘𝐹) ≠ ∅) ↔ ((𝜑 ∧ 𝑧 ∈ 𝐶) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴))‘𝐹) ≠ ∅)))
201	200, 86	chvarvv 1989	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴))‘𝐹) ≠ ∅)
202	42	hausflf2 23892	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐾 ∈ Haus ∧ (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴) ∈ (Fil‘𝐴) ∧ 𝐹:𝐴⟶𝐵) ∧ ((𝐾 fLimf (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴))‘𝐹) ≠ ∅) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴))‘𝐹) ≈ 1_o)
203	182, 190, 191, 201, 202	syl31anc 1375	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴))‘𝐹) ≈ 1_o)
204		en1b 8999	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐾 fLimf (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴))‘𝐹) ≈ 1_o ↔ ((𝐾 fLimf (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴))‘𝐹) = {∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴))‘𝐹)})
205	203, 204	sylib 218	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴))‘𝐹) = {∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴))‘𝐹)})
206	205	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐶) ∧ (𝑣 ∩ 𝐴) ∈ (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴)) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴))‘𝐹) = {∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴))‘𝐹)})
207	109	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → 𝐾 ∈ (TopOn‘𝐵))
208		flfval 23884	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐾 ∈ (TopOn‘𝐵) ∧ (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴) ∈ (Fil‘𝐴) ∧ 𝐹:𝐴⟶𝐵) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴))‘𝐹) = (𝐾 fLim ((𝐵 FilMap 𝐹)‘(((nei‘𝐽)‘{𝑧}) ↾_t 𝐴))))
209	207, 190, 191, 208	syl3anc 1373	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴))‘𝐹) = (𝐾 fLim ((𝐵 FilMap 𝐹)‘(((nei‘𝐽)‘{𝑧}) ↾_t 𝐴))))
210	209	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐶) ∧ (𝑣 ∩ 𝐴) ∈ (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴)) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴))‘𝐹) = (𝐾 fLim ((𝐵 FilMap 𝐹)‘(((nei‘𝐽)‘{𝑧}) ↾_t 𝐴))))
211	32	uniexd 7721	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ∪ 𝐾 ∈ V)
212	211	ad2antrr 726	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐶) ∧ (𝑣 ∩ 𝐴) ∈ (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴)) → ∪ 𝐾 ∈ V)
213	42, 212	eqeltrid 2833	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐶) ∧ (𝑣 ∩ 𝐴) ∈ (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴)) → 𝐵 ∈ V)
214	190	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐶) ∧ (𝑣 ∩ 𝐴) ∈ (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴)) → (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴) ∈ (Fil‘𝐴))
215		filfbas 23742	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((nei‘𝐽)‘{𝑧}) ↾_t 𝐴) ∈ (Fil‘𝐴) → (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴) ∈ (fBas‘𝐴))
216	214, 215	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐶) ∧ (𝑣 ∩ 𝐴) ∈ (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴)) → (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴) ∈ (fBas‘𝐴))
217	36	ad2antrr 726	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐶) ∧ (𝑣 ∩ 𝐴) ∈ (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴)) → 𝐹:𝐴⟶𝐵)
218		simpr 484	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐶) ∧ (𝑣 ∩ 𝐴) ∈ (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴)) → (𝑣 ∩ 𝐴) ∈ (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴))
219		fgfil 23769	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((nei‘𝐽)‘{𝑧}) ↾_t 𝐴) ∈ (Fil‘𝐴) → (𝐴filGen(((nei‘𝐽)‘{𝑧}) ↾_t 𝐴)) = (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴))
220	190, 219	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → (𝐴filGen(((nei‘𝐽)‘{𝑧}) ↾_t 𝐴)) = (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴))
221	220	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐶) ∧ (𝑣 ∩ 𝐴) ∈ (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴)) → (𝐴filGen(((nei‘𝐽)‘{𝑧}) ↾_t 𝐴)) = (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴))
222	218, 221	eleqtrrd 2832	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐶) ∧ (𝑣 ∩ 𝐴) ∈ (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴)) → (𝑣 ∩ 𝐴) ∈ (𝐴filGen(((nei‘𝐽)‘{𝑧}) ↾_t 𝐴)))
223		eqid 2730	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐴filGen(((nei‘𝐽)‘{𝑧}) ↾_t 𝐴)) = (𝐴filGen(((nei‘𝐽)‘{𝑧}) ↾_t 𝐴))
224	223	imaelfm 23845	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐵 ∈ V ∧ (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴) ∈ (fBas‘𝐴) ∧ 𝐹:𝐴⟶𝐵) ∧ (𝑣 ∩ 𝐴) ∈ (𝐴filGen(((nei‘𝐽)‘{𝑧}) ↾_t 𝐴))) → (𝐹 “ (𝑣 ∩ 𝐴)) ∈ ((𝐵 FilMap 𝐹)‘(((nei‘𝐽)‘{𝑧}) ↾_t 𝐴)))
225	213, 216, 217, 222, 224	syl31anc 1375	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐶) ∧ (𝑣 ∩ 𝐴) ∈ (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴)) → (𝐹 “ (𝑣 ∩ 𝐴)) ∈ ((𝐵 FilMap 𝐹)‘(((nei‘𝐽)‘{𝑧}) ↾_t 𝐴)))
226		flimclsi 23872	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹 “ (𝑣 ∩ 𝐴)) ∈ ((𝐵 FilMap 𝐹)‘(((nei‘𝐽)‘{𝑧}) ↾_t 𝐴)) → (𝐾 fLim ((𝐵 FilMap 𝐹)‘(((nei‘𝐽)‘{𝑧}) ↾_t 𝐴))) ⊆ ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))))
227	225, 226	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐶) ∧ (𝑣 ∩ 𝐴) ∈ (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴)) → (𝐾 fLim ((𝐵 FilMap 𝐹)‘(((nei‘𝐽)‘{𝑧}) ↾_t 𝐴))) ⊆ ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))))
228	210, 227	eqsstrd 3984	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐶) ∧ (𝑣 ∩ 𝐴) ∈ (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴)) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴))‘𝐹) ⊆ ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))))
229	206, 228	eqsstrrd 3985	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐶) ∧ (𝑣 ∩ 𝐴) ∈ (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴)) → {∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴))‘𝐹)} ⊆ ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))))
230		fvex 6874	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐾 fLimf (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴))‘𝐹) ∈ V
231	230	uniex 7720	. . . . . . . . . . . . . . . 16 ⊢ ∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴))‘𝐹) ∈ V
232	231	snss 4752	. . . . . . . . . . . . . . 15 ⊢ (∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴))‘𝐹) ∈ ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))) ↔ {∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴))‘𝐹)} ⊆ ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))))
233	229, 232	sylibr 234	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐶) ∧ (𝑣 ∩ 𝐴) ∈ (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴)) → ∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴))‘𝐹) ∈ ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))))
234	181, 233	eqeltrd 2829	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐶) ∧ (𝑣 ∩ 𝐴) ∈ (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴)) → (((𝐽CnExt𝐾)‘𝐹)‘𝑧) ∈ ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))))
235	166, 172, 179, 234	syl21anc 837	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ 𝑧 ∈ 𝑣) → (((𝐽CnExt𝐾)‘𝐹)‘𝑧) ∈ ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))))
236	235	adantlr 715	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))) ⊆ 𝑤) ∧ 𝑧 ∈ 𝑣) → (((𝐽CnExt𝐾)‘𝐹)‘𝑧) ∈ ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))))
237	165, 236	sseldd 3950	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))) ⊆ 𝑤) ∧ 𝑧 ∈ 𝑣) → (((𝐽CnExt𝐾)‘𝐹)‘𝑧) ∈ 𝑤)
238	237	ralrimiva 3126	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))) ⊆ 𝑤) → ∀𝑧 ∈ 𝑣 (((𝐽CnExt𝐾)‘𝐹)‘𝑧) ∈ 𝑤)
239	238	expl 457	. . . . . . . 8 ⊢ (𝜑 → ((𝑣 ∈ 𝐽 ∧ ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))) ⊆ 𝑤) → ∀𝑧 ∈ 𝑣 (((𝐽CnExt𝐾)‘𝐹)‘𝑧) ∈ 𝑤))
240	239	reximdv 3149	. . . . . . 7 ⊢ (𝜑 → (∃𝑣 ∈ ((nei‘𝐽)‘{𝑥})(𝑣 ∈ 𝐽 ∧ ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))) ⊆ 𝑤) → ∃𝑣 ∈ ((nei‘𝐽)‘{𝑥})∀𝑧 ∈ 𝑣 (((𝐽CnExt𝐾)‘𝐹)‘𝑧) ∈ 𝑤))
241	240	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) → (∃𝑣 ∈ ((nei‘𝐽)‘{𝑥})(𝑣 ∈ 𝐽 ∧ ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))) ⊆ 𝑤) → ∃𝑣 ∈ ((nei‘𝐽)‘{𝑥})∀𝑧 ∈ 𝑣 (((𝐽CnExt𝐾)‘𝐹)‘𝑧) ∈ 𝑤))
242	164, 241	mpd 15	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) → ∃𝑣 ∈ ((nei‘𝐽)‘{𝑥})∀𝑧 ∈ 𝑣 (((𝐽CnExt𝐾)‘𝐹)‘𝑧) ∈ 𝑤)
243	65, 42, 4, 32, 36, 69, 76, 86	cnextf 23960	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐽CnExt𝐾)‘𝐹):𝐶⟶𝐵)
244	243	ffund 6695	. . . . . . . . 9 ⊢ (𝜑 → Fun ((𝐽CnExt𝐾)‘𝐹))
245	244	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ ((nei‘𝐽)‘{𝑥})) → Fun ((𝐽CnExt𝐾)‘𝐹))
246	65	neii1 23000	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝑣 ∈ ((nei‘𝐽)‘{𝑥})) → 𝑣 ⊆ 𝐶)
247	4, 246	sylan 580	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ ((nei‘𝐽)‘{𝑥})) → 𝑣 ⊆ 𝐶)
248	243	fdmd 6701	. . . . . . . . . 10 ⊢ (𝜑 → dom ((𝐽CnExt𝐾)‘𝐹) = 𝐶)
249	248	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ ((nei‘𝐽)‘{𝑥})) → dom ((𝐽CnExt𝐾)‘𝐹) = 𝐶)
250	247, 249	sseqtrrd 3987	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ ((nei‘𝐽)‘{𝑥})) → 𝑣 ⊆ dom ((𝐽CnExt𝐾)‘𝐹))
251		funimass4 6928	. . . . . . . 8 ⊢ ((Fun ((𝐽CnExt𝐾)‘𝐹) ∧ 𝑣 ⊆ dom ((𝐽CnExt𝐾)‘𝐹)) → ((((𝐽CnExt𝐾)‘𝐹) “ 𝑣) ⊆ 𝑤 ↔ ∀𝑧 ∈ 𝑣 (((𝐽CnExt𝐾)‘𝐹)‘𝑧) ∈ 𝑤))
252	245, 250, 251	syl2anc 584	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ ((nei‘𝐽)‘{𝑥})) → ((((𝐽CnExt𝐾)‘𝐹) “ 𝑣) ⊆ 𝑤 ↔ ∀𝑧 ∈ 𝑣 (((𝐽CnExt𝐾)‘𝐹)‘𝑧) ∈ 𝑤))
253	252	biimprd 248	. . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ ((nei‘𝐽)‘{𝑥})) → (∀𝑧 ∈ 𝑣 (((𝐽CnExt𝐾)‘𝐹)‘𝑧) ∈ 𝑤 → (((𝐽CnExt𝐾)‘𝐹) “ 𝑣) ⊆ 𝑤))
254	253	reximdva 3147	. . . . 5 ⊢ (𝜑 → (∃𝑣 ∈ ((nei‘𝐽)‘{𝑥})∀𝑧 ∈ 𝑣 (((𝐽CnExt𝐾)‘𝐹)‘𝑧) ∈ 𝑤 → ∃𝑣 ∈ ((nei‘𝐽)‘{𝑥})(((𝐽CnExt𝐾)‘𝐹) “ 𝑣) ⊆ 𝑤))
255	1, 242, 254	sylc 65	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) → ∃𝑣 ∈ ((nei‘𝐽)‘{𝑥})(((𝐽CnExt𝐾)‘𝐹) “ 𝑣) ⊆ 𝑤)
256	255	ralrimiva 3126	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ∀𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})∃𝑣 ∈ ((nei‘𝐽)‘{𝑥})(((𝐽CnExt𝐾)‘𝐹) “ 𝑣) ⊆ 𝑤)
257	256	ralrimiva 3126	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐶 ∀𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})∃𝑣 ∈ ((nei‘𝐽)‘{𝑥})(((𝐽CnExt𝐾)‘𝐹) “ 𝑣) ⊆ 𝑤)
258	65, 42	cnnei 23176	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ ((𝐽CnExt𝐾)‘𝐹):𝐶⟶𝐵) → (((𝐽CnExt𝐾)‘𝐹) ∈ (𝐽 Cn 𝐾) ↔ ∀𝑥 ∈ 𝐶 ∀𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})∃𝑣 ∈ ((nei‘𝐽)‘{𝑥})(((𝐽CnExt𝐾)‘𝐹) “ 𝑣) ⊆ 𝑤))
259	4, 34, 243, 258	syl3anc 1373	. 2 ⊢ (𝜑 → (((𝐽CnExt𝐾)‘𝐹) ∈ (𝐽 Cn 𝐾) ↔ ∀𝑥 ∈ 𝐶 ∀𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})∃𝑣 ∈ ((nei‘𝐽)‘{𝑥})(((𝐽CnExt𝐾)‘𝐹) “ 𝑣) ⊆ 𝑤))
260	257, 259	mpbird 257	1 ⊢ (𝜑 → ((𝐽CnExt𝐾)‘𝐹) ∈ (𝐽 Cn 𝐾))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ≠ wne 2926 ∀wral 3045 ∃wrex 3054 Vcvv 3450 ∩ cin 3916 ⊆ wss 3917 ∅c0 4299 {csn 4592 ∪ cuni 4874 class class class wbr 5110 dom cdm 5641 ran crn 5642 “ cima 5644 Fun wfun 6508 ⟶wf 6510 ‘cfv 6514 (class class class)co 7390 1_oc1o 8430 ≈ cen 8918 ↾_t crest 17390 fBascfbas 21259 filGencfg 21260 Topctop 22787 TopOnctopon 22804 clsccl 22912 neicnei 22991 Cn ccn 23118 Hauscha 23202 Regcreg 23203 Filcfil 23739 FilMap cfm 23827 fLim cflim 23828 fLimf cflf 23829 CnExtccnext 23953

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 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714

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 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-int 4914 df-iun 4960 df-iin 4961 df-br 5111 df-opab 5173 df-mpt 5192 df-id 5536 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-ov 7393 df-oprab 7394 df-mpo 7395 df-1st 7971 df-2nd 7972 df-1o 8437 df-map 8804 df-pm 8805 df-en 8922 df-rest 17392 df-topgen 17413 df-fbas 21268 df-fg 21269 df-top 22788 df-topon 22805 df-cld 22913 df-ntr 22914 df-cls 22915 df-nei 22992 df-cn 23121 df-cnp 23122 df-haus 23209 df-reg 23210 df-fil 23740 df-fm 23832 df-flim 23833 df-flf 23834 df-cnext 23954

This theorem is referenced by: cnextucn 24197

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