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Theorem cnextcn 22677

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 765	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) → 𝜑)
2		simpll 765	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ (𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)}) ∧ 𝑑 ∈ ((nei‘𝐽)‘{𝑥}) ∧ ((cls‘𝐾)‘(𝐹 “ (𝑑 ∩ 𝐴))) ⊆ 𝑤)) → 𝜑)
3		simpr3 1192	. . . . . . . . 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 724	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ (𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)}) ∧ 𝑑 ∈ ((nei‘𝐽)‘{𝑥}) ∧ ((cls‘𝐾)‘(𝐹 “ (𝑑 ∩ 𝐴))) ⊆ 𝑤)) → 𝐽 ∈ Top)
6		simpr2 1191	. . . . . . . . . 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 21718	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝑑 ∈ ((nei‘𝐽)‘{𝑥})) → ∃𝑣 ∈ 𝐽 ({𝑥} ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑑))
8	5, 6, 7	syl2anc 586	. . . . . . . . 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 3499	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑥 ∈ V
10	9	snss 4720	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ 𝑣 ↔ {𝑥} ⊆ 𝑣)
11	10	biimpri 230	. . . . . . . . . . . . . . . . . 18 ⊢ ({𝑥} ⊆ 𝑣 → 𝑥 ∈ 𝑣)
12	11	anim1i 616	. . . . . . . . . . . . . . . . 17 ⊢ (({𝑥} ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑑) → (𝑥 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑑))
13	12	anim2i 618	. . . . . . . . . . . . . . . 16 ⊢ ((𝑣 ∈ 𝐽 ∧ ({𝑥} ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑑)) → (𝑣 ∈ 𝐽 ∧ (𝑥 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑑)))
14	13	anim2i 618	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐽 ∧ ({𝑥} ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑑))) → (𝜑 ∧ (𝑣 ∈ 𝐽 ∧ (𝑥 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑑))))
15	14	ex 415	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑣 ∈ 𝐽 ∧ ({𝑥} ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑑)) → (𝜑 ∧ (𝑣 ∈ 𝐽 ∧ (𝑥 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑑)))))
16		3anass 1091	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐽 ∧ 𝑥 ∈ 𝑣) ↔ (𝜑 ∧ (𝑣 ∈ 𝐽 ∧ 𝑥 ∈ 𝑣)))
17	16	anbi1i 625	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽 ∧ 𝑥 ∈ 𝑣) ∧ 𝑣 ⊆ 𝑑) ↔ ((𝜑 ∧ (𝑣 ∈ 𝐽 ∧ 𝑥 ∈ 𝑣)) ∧ 𝑣 ⊆ 𝑑))
18		anass 471	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐽 ∧ 𝑥 ∈ 𝑣)) ∧ 𝑣 ⊆ 𝑑) ↔ (𝜑 ∧ ((𝑣 ∈ 𝐽 ∧ 𝑥 ∈ 𝑣) ∧ 𝑣 ⊆ 𝑑)))
19		anass 471	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑣 ∈ 𝐽 ∧ 𝑥 ∈ 𝑣) ∧ 𝑣 ⊆ 𝑑) ↔ (𝑣 ∈ 𝐽 ∧ (𝑥 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑑)))
20	19	anbi2i 624	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑣 ∈ 𝐽 ∧ 𝑥 ∈ 𝑣) ∧ 𝑣 ⊆ 𝑑)) ↔ (𝜑 ∧ (𝑣 ∈ 𝐽 ∧ (𝑥 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑑))))
21	17, 18, 20	3bitri 299	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽 ∧ 𝑥 ∈ 𝑣) ∧ 𝑣 ⊆ 𝑑) ↔ (𝜑 ∧ (𝑣 ∈ 𝐽 ∧ (𝑥 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑑))))
22		opnneip 21729	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐽 ∈ Top ∧ 𝑣 ∈ 𝐽 ∧ 𝑥 ∈ 𝑣) → 𝑣 ∈ ((nei‘𝐽)‘{𝑥}))
23	4, 22	syl3an1 1159	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐽 ∧ 𝑥 ∈ 𝑣) → 𝑣 ∈ ((nei‘𝐽)‘{𝑥}))
24	23	adantr 483	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽 ∧ 𝑥 ∈ 𝑣) ∧ 𝑣 ⊆ 𝑑) → 𝑣 ∈ ((nei‘𝐽)‘{𝑥}))
25		simpr2 1191	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑣 ⊆ 𝑑 ∧ (𝜑 ∧ 𝑣 ∈ 𝐽 ∧ 𝑥 ∈ 𝑣)) → 𝑣 ∈ 𝐽)
26	25	ex 415	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 ⊆ 𝑑 → ((𝜑 ∧ 𝑣 ∈ 𝐽 ∧ 𝑥 ∈ 𝑣) → 𝑣 ∈ 𝐽))
27	26	imdistanri 572	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽 ∧ 𝑥 ∈ 𝑣) ∧ 𝑣 ⊆ 𝑑) → (𝑣 ∈ 𝐽 ∧ 𝑣 ⊆ 𝑑))
28	24, 27	jca 514	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽 ∧ 𝑥 ∈ 𝑣) ∧ 𝑣 ⊆ 𝑑) → (𝑣 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝑣 ∈ 𝐽 ∧ 𝑣 ⊆ 𝑑)))
29	21, 28	sylbir 237	. . . . . . . . . . . . . 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 483	. . . . . . . . . . . 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 21941	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐾 ∈ Haus → 𝐾 ∈ Top)
34	32, 33	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐾 ∈ Top)
35		imassrn 5942	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐹 “ (𝑑 ∩ 𝐴)) ⊆ ran 𝐹
36		cnextf.5	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
37	36	frnd 6523	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ran 𝐹 ⊆ 𝐵)
38	35, 37	sstrid 3980	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐹 “ (𝑑 ∩ 𝐴)) ⊆ 𝐵)
39		ssrin 4212	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑣 ⊆ 𝑑 → (𝑣 ∩ 𝐴) ⊆ (𝑑 ∩ 𝐴))
40		imass2 5967	. . . . . . . . . . . . . . . . . . 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 21664	. . . . . . . . . . . . . . . . . 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 1418	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑣 ⊆ 𝑑) → ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))) ⊆ ((cls‘𝐾)‘(𝐹 “ (𝑑 ∩ 𝐴))))
45		sstr 3977	. . . . . . . . . . . . . . . . 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 582	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑣 ⊆ 𝑑) ∧ ((cls‘𝐾)‘(𝐹 “ (𝑑 ∩ 𝐴))) ⊆ 𝑤) → ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))) ⊆ 𝑤)
47	46	an32s 650	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ((cls‘𝐾)‘(𝐹 “ (𝑑 ∩ 𝐴))) ⊆ 𝑤) ∧ 𝑣 ⊆ 𝑑) → ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))) ⊆ 𝑤)
48	47	ex 415	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((cls‘𝐾)‘(𝐹 “ (𝑑 ∩ 𝐴))) ⊆ 𝑤) → (𝑣 ⊆ 𝑑 → ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))) ⊆ 𝑤))
49	48	anim2d 613	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((cls‘𝐾)‘(𝐹 “ (𝑑 ∩ 𝐴))) ⊆ 𝑤) → ((𝑣 ∈ 𝐽 ∧ 𝑣 ⊆ 𝑑) → (𝑣 ∈ 𝐽 ∧ ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))) ⊆ 𝑤)))
50	49	anim2d 613	. . . . . . . . . . . 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 3273	. . . . . . . . . 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 409	. . . . . . . . 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 835	. . . . . . . 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 1356	. . . . . . 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 487	. . . . . . . . 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 781	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) ∧ 𝑢 ∈ (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴)) ∧ ((cls‘𝐾)‘(𝐹 “ 𝑢)) ⊆ 𝑤) → 𝜑)
58		simplr 767	. . . . . . . . 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 5927	. . . . . . . . . . . . . 14 ⊢ (𝑢 = (𝑑 ∩ 𝐴) → (𝐹 “ 𝑢) = (𝐹 “ (𝑑 ∩ 𝐴)))
60	59	fveq2d 6676	. . . . . . . . . . . . 13 ⊢ (𝑢 = (𝑑 ∩ 𝐴) → ((cls‘𝐾)‘(𝐹 “ 𝑢)) = ((cls‘𝐾)‘(𝐹 “ (𝑑 ∩ 𝐴))))
61	60	sseq1d 4000	. . . . . . . . . . . 12 ⊢ (𝑢 = (𝑑 ∩ 𝐴) → (((cls‘𝐾)‘(𝐹 “ 𝑢)) ⊆ 𝑤 ↔ ((cls‘𝐾)‘(𝐹 “ (𝑑 ∩ 𝐴))) ⊆ 𝑤))
62	61	biimpcd 251	. . . . . . . . . . 11 ⊢ (((cls‘𝐾)‘(𝐹 “ 𝑢)) ⊆ 𝑤 → (𝑢 = (𝑑 ∩ 𝐴) → ((cls‘𝐾)‘(𝐹 “ (𝑑 ∩ 𝐴))) ⊆ 𝑤))
63	62	reximdv 3275	. . . . . . . . . 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 6687	. . . . . . . . . . . 12 ⊢ (𝜑 → ((nei‘𝐽)‘{𝑥}) ∈ V)
65		cnextf.1	. . . . . . . . . . . . . . . 16 ⊢ 𝐶 = ∪ 𝐽
66	65	toptopon 21527	. . . . . . . . . . . . . . 15 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝐶))
67	4, 66	sylib 220	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝐶))
68	67	elfvexd 6706	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐶 ∈ V)
69		cnextf.a	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 ⊆ 𝐶)
70	68, 69	ssexd 5230	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ V)
71		elrest 16703	. . . . . . . . . . . 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 586	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑢 ∈ (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴) ↔ ∃𝑑 ∈ ((nei‘𝐽)‘{𝑥})𝑢 = (𝑑 ∩ 𝐴)))
73	72	biimpa 479	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑢 ∈ (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴)) → ∃𝑑 ∈ ((nei‘𝐽)‘{𝑥})𝑢 = (𝑑 ∩ 𝐴))
74	63, 73	impel 508	. . . . . . . . 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 834	. . . . . . . 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 2897	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐶 ↔ 𝑦 ∈ 𝐶))
78	77	anbi2d 630	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑦 → ((𝜑 ∧ 𝑥 ∈ 𝐶) ↔ (𝜑 ∧ 𝑦 ∈ 𝐶)))
79		sneq 4579	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦})
80	79	fveq2d 6676	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑦 → ((nei‘𝐽)‘{𝑥}) = ((nei‘𝐽)‘{𝑦}))
81	80	oveq1d 7173	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = 𝑦 → (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴) = (((nei‘𝐽)‘{𝑦}) ↾_t 𝐴))
82	81	oveq2d 7174	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑦 → (𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴)) = (𝐾 fLimf (((nei‘𝐽)‘{𝑦}) ↾_t 𝐴)))
83	82	fveq1d 6674	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑦 → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴))‘𝐹) = ((𝐾 fLimf (((nei‘𝐽)‘{𝑦}) ↾_t 𝐴))‘𝐹))
84	83	neeq1d 3077	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑦 → (((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴))‘𝐹) ≠ ∅ ↔ ((𝐾 fLimf (((nei‘𝐽)‘{𝑦}) ↾_t 𝐴))‘𝐹) ≠ ∅))
85	78, 84	imbi12d 347	. . . . . . . . . . . . . . . . 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 2005	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑦}) ↾_t 𝐴))‘𝐹) ≠ ∅)
88	65, 42, 4, 32, 36, 69, 76, 87	cnextfvval 22675	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (((𝐽CnExt𝐾)‘𝐹)‘𝑥) = ∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴))‘𝐹))
89		fvex 6685	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴))‘𝐹) ∈ V
90	89	uniex 7469	. . . . . . . . . . . . . . . . 17 ⊢ ∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴))‘𝐹) ∈ V
91	90	snid 4603	. . . . . . . . . . . . . . . 16 ⊢ ∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴))‘𝐹) ∈ {∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴))‘𝐹)}
92	32	adantr 483	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐾 ∈ Haus)
93	76	eleq2d 2900	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↔ 𝑥 ∈ 𝐶))
94	93	biimpar 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝑥 ∈ ((cls‘𝐽)‘𝐴))
95	67	adantr 483	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐽 ∈ (TopOn‘𝐶))
96	69	adantr 483	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐴 ⊆ 𝐶)
97		simpr 487	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝑥 ∈ 𝐶)
98		trnei 22502	. . . . . . . . . . . . . . . . . . . 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 1367	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↔ (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴) ∈ (Fil‘𝐴)))
100	94, 99	mpbid 234	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴) ∈ (Fil‘𝐴))
101	36	adantr 483	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐹:𝐴⟶𝐵)
102	42	hausflf2 22608	. . . . . . . . . . . . . . . . . 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 1369	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴))‘𝐹) ≈ 1_o)
104		en1b 8579	. . . . . . . . . . . . . . . . 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 220	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴))‘𝐹) = {∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴))‘𝐹)})
106	91, 105	eleqtrrid 2922	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴))‘𝐹) ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴))‘𝐹))
107	88, 106	eqeltrd 2915	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴))‘𝐹))
108	42	toptopon 21527	. . . . . . . . . . . . . . . . 17 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝐵))
109	34, 108	sylib 220	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝐵))
110	109	adantr 483	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐾 ∈ (TopOn‘𝐵))
111		flfnei 22601	. . . . . . . . . . . . . . 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 1367	. . . . . . . . . . . . . 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 234	. . . . . . . . . . . . 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 498	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ∀𝑏 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})∃𝑢 ∈ (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴)(𝐹 “ 𝑢) ⊆ 𝑏)
115	114	r19.21bi 3210	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑏 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) → ∃𝑢 ∈ (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴)(𝐹 “ 𝑢) ⊆ 𝑏)
116	115	ad4ant13 749	. . . . . . . . . 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 728	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑏 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑤) → 𝐾 ∈ Top)
118		simplr 767	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑏 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑤) → 𝑏 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)}))
119	42	neii1 21716	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ Top ∧ 𝑏 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) → 𝑏 ⊆ 𝐵)
120	117, 118, 119	syl2anc 586	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑏 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑤) → 𝑏 ⊆ 𝐵)
121		simpr 487	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑏 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑤) → ((cls‘𝐾)‘𝑏) ⊆ 𝑤)
122	42	clsss 21664	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ Top ∧ 𝑏 ⊆ 𝐵 ∧ (𝐹 “ 𝑢) ⊆ 𝑏) → ((cls‘𝐾)‘(𝐹 “ 𝑢)) ⊆ ((cls‘𝐾)‘𝑏))
123		sstr 3977	. . . . . . . . . . . . . . . 16 ⊢ ((((cls‘𝐾)‘(𝐹 “ 𝑢)) ⊆ ((cls‘𝐾)‘𝑏) ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑤) → ((cls‘𝐾)‘(𝐹 “ 𝑢)) ⊆ 𝑤)
124	122, 123	sylan 582	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ Top ∧ 𝑏 ⊆ 𝐵 ∧ (𝐹 “ 𝑢) ⊆ 𝑏) ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑤) → ((cls‘𝐾)‘(𝐹 “ 𝑢)) ⊆ 𝑤)
125	124	3an1rs 1355	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ Top ∧ 𝑏 ⊆ 𝐵 ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑤) ∧ (𝐹 “ 𝑢) ⊆ 𝑏) → ((cls‘𝐾)‘(𝐹 “ 𝑢)) ⊆ 𝑤)
126	125	ex 415	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ Top ∧ 𝑏 ⊆ 𝐵 ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑤) → ((𝐹 “ 𝑢) ⊆ 𝑏 → ((cls‘𝐾)‘(𝐹 “ 𝑢)) ⊆ 𝑤))
127	126	reximdv 3275	. . . . . . . . . . . 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 1367	. . . . . . . . . . 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 717	. . . . . . . . . 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 724	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) → 𝐾 ∈ Top)
132		cnextcn.8	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐾 ∈ Reg)
133	132	ad2antrr 724	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) → 𝐾 ∈ Reg)
134	133	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) ∧ 𝑐 ∈ 𝐾) ∧ ((((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑐 ∧ 𝑐 ⊆ 𝑤)) → 𝐾 ∈ Reg)
135		simplr 767	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) ∧ 𝑐 ∈ 𝐾) ∧ ((((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑐 ∧ 𝑐 ⊆ 𝑤)) → 𝑐 ∈ 𝐾)
136		simprl 769	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) ∧ 𝑐 ∈ 𝐾) ∧ ((((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑐 ∧ 𝑐 ⊆ 𝑤)) → (((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑐)
137		regsep 21944	. . . . . . . . . . . . 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 1367	. . . . . . . . . . . 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 3977	. . . . . . . . . . . . . . . 16 ⊢ ((((cls‘𝐾)‘𝑏) ⊆ 𝑐 ∧ 𝑐 ⊆ 𝑤) → ((cls‘𝐾)‘𝑏) ⊆ 𝑤)
140	139	expcom 416	. . . . . . . . . . . . . . 15 ⊢ (𝑐 ⊆ 𝑤 → (((cls‘𝐾)‘𝑏) ⊆ 𝑐 → ((cls‘𝐾)‘𝑏) ⊆ 𝑤))
141	140	anim2d 613	. . . . . . . . . . . . . 14 ⊢ (𝑐 ⊆ 𝑤 → (((((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑏 ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑐) → ((((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑏 ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑤)))
142	141	reximdv 3275	. . . . . . . . . . . . 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 727	. . . . . . . . . . . 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 487	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) → 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)}))
146		neii2 21718	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ Top ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) → ∃𝑐 ∈ 𝐾 ({(((𝐽CnExt𝐾)‘𝐹)‘𝑥)} ⊆ 𝑐 ∧ 𝑐 ⊆ 𝑤))
147		fvex 6685	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ V
148	147	snss 4720	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑐 ↔ {(((𝐽CnExt𝐾)‘𝐹)‘𝑥)} ⊆ 𝑐)
149	148	anbi1i 625	. . . . . . . . . . . . . . 15 ⊢ (((((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑐 ∧ 𝑐 ⊆ 𝑤) ↔ ({(((𝐽CnExt𝐾)‘𝐹)‘𝑥)} ⊆ 𝑐 ∧ 𝑐 ⊆ 𝑤))
150	149	biimpri 230	. . . . . . . . . . . . . 14 ⊢ (({(((𝐽CnExt𝐾)‘𝐹)‘𝑥)} ⊆ 𝑐 ∧ 𝑐 ⊆ 𝑤) → ((((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑐 ∧ 𝑐 ⊆ 𝑤))
151	150	reximi 3245	. . . . . . . . . . . . 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 586	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) → ∃𝑐 ∈ 𝐾 ((((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑐 ∧ 𝑐 ⊆ 𝑤))
154	144, 153	r19.29a 3291	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) → ∃𝑏 ∈ 𝐾 ((((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑏 ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑤))
155		anass 471	. . . . . . . . . . . 12 ⊢ (((𝑏 ∈ 𝐾 ∧ (((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑏) ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑤) ↔ (𝑏 ∈ 𝐾 ∧ ((((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑏 ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑤)))
156		opnneip 21729	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ Top ∧ 𝑏 ∈ 𝐾 ∧ (((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑏) → 𝑏 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)}))
157	156	3expib 1118	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ Top → ((𝑏 ∈ 𝐾 ∧ (((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑏) → 𝑏 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})))
158	157	anim1d 612	. . . . . . . . . . . 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	syl5bir 245	. . . . . . . . . . 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 3273	. . . . . . . . . 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 3291	. . . . . . . 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 3291	. . . . . . 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 3291	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) → ∃𝑣 ∈ ((nei‘𝐽)‘{𝑥})(𝑣 ∈ 𝐽 ∧ ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))) ⊆ 𝑤))
165		simplr 767	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))) ⊆ 𝑤) ∧ 𝑧 ∈ 𝑣) → ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))) ⊆ 𝑤)
166		simpll 765	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ 𝑧 ∈ 𝑣) → 𝜑)
167	4	ad2antrr 724	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ 𝑧 ∈ 𝑣) → 𝐽 ∈ Top)
168		simplr 767	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ 𝑧 ∈ 𝑣) → 𝑣 ∈ 𝐽)
169	65	eltopss 21517	. . . . . . . . . . . . . . 15 ⊢ ((𝐽 ∈ Top ∧ 𝑣 ∈ 𝐽) → 𝑣 ⊆ 𝐶)
170	167, 168, 169	syl2anc 586	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ 𝑧 ∈ 𝑣) → 𝑣 ⊆ 𝐶)
171		simpr 487	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ 𝑧 ∈ 𝑣) → 𝑧 ∈ 𝑣)
172	170, 171	sseldd 3970	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ 𝑧 ∈ 𝑣) → 𝑧 ∈ 𝐶)
173		fvexd 6687	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ 𝑧 ∈ 𝑣) → ((nei‘𝐽)‘{𝑧}) ∈ V)
174	70	ad2antrr 724	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ 𝑧 ∈ 𝑣) → 𝐴 ∈ V)
175		opnneip 21729	. . . . . . . . . . . . . . . 16 ⊢ ((𝐽 ∈ Top ∧ 𝑣 ∈ 𝐽 ∧ 𝑧 ∈ 𝑣) → 𝑣 ∈ ((nei‘𝐽)‘{𝑧}))
176	4, 175	syl3an1 1159	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐽 ∧ 𝑧 ∈ 𝑣) → 𝑣 ∈ ((nei‘𝐽)‘{𝑧}))
177	176	3expa 1114	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ 𝑧 ∈ 𝑣) → 𝑣 ∈ ((nei‘𝐽)‘{𝑧}))
178		elrestr 16704	. . . . . . . . . . . . . 14 ⊢ ((((nei‘𝐽)‘{𝑧}) ∈ V ∧ 𝐴 ∈ V ∧ 𝑣 ∈ ((nei‘𝐽)‘{𝑧})) → (𝑣 ∩ 𝐴) ∈ (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴))
179	173, 174, 177, 178	syl3anc 1367	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ 𝑧 ∈ 𝑣) → (𝑣 ∩ 𝐴) ∈ (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴))
180	65, 42, 4, 32, 36, 69, 76, 86	cnextfvval 22675	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → (((𝐽CnExt𝐾)‘𝐹)‘𝑧) = ∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴))‘𝐹))
181	180	adantr 483	. . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → 𝐾 ∈ Haus)
183	76	eleq2d 2900	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝑧 ∈ ((cls‘𝐽)‘𝐴) ↔ 𝑧 ∈ 𝐶))
184	183	biimpar 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → 𝑧 ∈ ((cls‘𝐽)‘𝐴))
185	67	adantr 483	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → 𝐽 ∈ (TopOn‘𝐶))
186	69	adantr 483	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → 𝐴 ⊆ 𝐶)
187		simpr 487	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → 𝑧 ∈ 𝐶)
188		trnei 22502	. . . . . . . . . . . . . . . . . . . . 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 1367	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → (𝑧 ∈ ((cls‘𝐽)‘𝐴) ↔ (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴) ∈ (Fil‘𝐴)))
190	184, 189	mpbid 234	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴) ∈ (Fil‘𝐴))
191	36	adantr 483	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → 𝐹:𝐴⟶𝐵)
192		eleq1w 2897	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐶 ↔ 𝑧 ∈ 𝐶))
193	192	anbi2d 630	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = 𝑧 → ((𝜑 ∧ 𝑥 ∈ 𝐶) ↔ (𝜑 ∧ 𝑧 ∈ 𝐶)))
194		sneq 4579	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑥 = 𝑧 → {𝑥} = {𝑧})
195	194	fveq2d 6676	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥 = 𝑧 → ((nei‘𝐽)‘{𝑥}) = ((nei‘𝐽)‘{𝑧}))
196	195	oveq1d 7173	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 = 𝑧 → (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴) = (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴))
197	196	oveq2d 7174	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = 𝑧 → (𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴)) = (𝐾 fLimf (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴)))
198	197	fveq1d 6674	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑧 → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴))‘𝐹) = ((𝐾 fLimf (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴))‘𝐹))
199	198	neeq1d 3077	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = 𝑧 → (((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴))‘𝐹) ≠ ∅ ↔ ((𝐾 fLimf (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴))‘𝐹) ≠ ∅))
200	193, 199	imbi12d 347	. . . . . . . . . . . . . . . . . . . 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 2005	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴))‘𝐹) ≠ ∅)
202	42	hausflf2 22608	. . . . . . . . . . . . . . . . . . 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 1369	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴))‘𝐹) ≈ 1_o)
204		en1b 8579	. . . . . . . . . . . . . . . . . 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 220	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴))‘𝐹) = {∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴))‘𝐹)})
206	205	adantr 483	. . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → 𝐾 ∈ (TopOn‘𝐵))
208		flfval 22600	. . . . . . . . . . . . . . . . . . 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 1367	. . . . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . . . 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 7470	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ∪ 𝐾 ∈ V)
212	211	ad2antrr 724	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐶) ∧ (𝑣 ∩ 𝐴) ∈ (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴)) → ∪ 𝐾 ∈ V)
213	42, 212	eqeltrid 2919	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐶) ∧ (𝑣 ∩ 𝐴) ∈ (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴)) → 𝐵 ∈ V)
214	190	adantr 483	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐶) ∧ (𝑣 ∩ 𝐴) ∈ (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴)) → (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴) ∈ (Fil‘𝐴))
215		filfbas 22458	. . . . . . . . . . . . . . . . . . . 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 724	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐶) ∧ (𝑣 ∩ 𝐴) ∈ (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴)) → 𝐹:𝐴⟶𝐵)
218		simpr 487	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐶) ∧ (𝑣 ∩ 𝐴) ∈ (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴)) → (𝑣 ∩ 𝐴) ∈ (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴))
219		fgfil 22485	. . . . . . . . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . . . . . . 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 2918	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐶) ∧ (𝑣 ∩ 𝐴) ∈ (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴)) → (𝑣 ∩ 𝐴) ∈ (𝐴filGen(((nei‘𝐽)‘{𝑧}) ↾_t 𝐴)))
223		eqid 2823	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐴filGen(((nei‘𝐽)‘{𝑧}) ↾_t 𝐴)) = (𝐴filGen(((nei‘𝐽)‘{𝑧}) ↾_t 𝐴))
224	223	imaelfm 22561	. . . . . . . . . . . . . . . . . . 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 1369	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐶) ∧ (𝑣 ∩ 𝐴) ∈ (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴)) → (𝐹 “ (𝑣 ∩ 𝐴)) ∈ ((𝐵 FilMap 𝐹)‘(((nei‘𝐽)‘{𝑧}) ↾_t 𝐴)))
226		flimclsi 22588	. . . . . . . . . . . . . . . . . 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 4007	. . . . . . . . . . . . . . . 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 4008	. . . . . . . . . . . . . . 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 6685	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐾 fLimf (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴))‘𝐹) ∈ V
231	230	uniex 7469	. . . . . . . . . . . . . . . 16 ⊢ ∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴))‘𝐹) ∈ V
232	231	snss 4720	. . . . . . . . . . . . . . 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 236	. . . . . . . . . . . . . 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 2915	. . . . . . . . . . . . 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 835	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ 𝑧 ∈ 𝑣) → (((𝐽CnExt𝐾)‘𝐹)‘𝑧) ∈ ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))))
236	235	adantlr 713	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))) ⊆ 𝑤) ∧ 𝑧 ∈ 𝑣) → (((𝐽CnExt𝐾)‘𝐹)‘𝑧) ∈ ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))))
237	165, 236	sseldd 3970	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))) ⊆ 𝑤) ∧ 𝑧 ∈ 𝑣) → (((𝐽CnExt𝐾)‘𝐹)‘𝑧) ∈ 𝑤)
238	237	ralrimiva 3184	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))) ⊆ 𝑤) → ∀𝑧 ∈ 𝑣 (((𝐽CnExt𝐾)‘𝐹)‘𝑧) ∈ 𝑤)
239	238	expl 460	. . . . . . . 8 ⊢ (𝜑 → ((𝑣 ∈ 𝐽 ∧ ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))) ⊆ 𝑤) → ∀𝑧 ∈ 𝑣 (((𝐽CnExt𝐾)‘𝐹)‘𝑧) ∈ 𝑤))
240	239	reximdv 3275	. . . . . . 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 724	. . . . . 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 22676	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐽CnExt𝐾)‘𝐹):𝐶⟶𝐵)
244	243	ffund 6520	. . . . . . . . 9 ⊢ (𝜑 → Fun ((𝐽CnExt𝐾)‘𝐹))
245	244	adantr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ ((nei‘𝐽)‘{𝑥})) → Fun ((𝐽CnExt𝐾)‘𝐹))
246	65	neii1 21716	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝑣 ∈ ((nei‘𝐽)‘{𝑥})) → 𝑣 ⊆ 𝐶)
247	4, 246	sylan 582	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ ((nei‘𝐽)‘{𝑥})) → 𝑣 ⊆ 𝐶)
248	243	fdmd 6525	. . . . . . . . . 10 ⊢ (𝜑 → dom ((𝐽CnExt𝐾)‘𝐹) = 𝐶)
249	248	adantr 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ ((nei‘𝐽)‘{𝑥})) → dom ((𝐽CnExt𝐾)‘𝐹) = 𝐶)
250	247, 249	sseqtrrd 4010	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ ((nei‘𝐽)‘{𝑥})) → 𝑣 ⊆ dom ((𝐽CnExt𝐾)‘𝐹))
251		funimass4 6732	. . . . . . . 8 ⊢ ((Fun ((𝐽CnExt𝐾)‘𝐹) ∧ 𝑣 ⊆ dom ((𝐽CnExt𝐾)‘𝐹)) → ((((𝐽CnExt𝐾)‘𝐹) “ 𝑣) ⊆ 𝑤 ↔ ∀𝑧 ∈ 𝑣 (((𝐽CnExt𝐾)‘𝐹)‘𝑧) ∈ 𝑤))
252	245, 250, 251	syl2anc 586	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ ((nei‘𝐽)‘{𝑥})) → ((((𝐽CnExt𝐾)‘𝐹) “ 𝑣) ⊆ 𝑤 ↔ ∀𝑧 ∈ 𝑣 (((𝐽CnExt𝐾)‘𝐹)‘𝑧) ∈ 𝑤))
253	252	biimprd 250	. . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ ((nei‘𝐽)‘{𝑥})) → (∀𝑧 ∈ 𝑣 (((𝐽CnExt𝐾)‘𝐹)‘𝑧) ∈ 𝑤 → (((𝐽CnExt𝐾)‘𝐹) “ 𝑣) ⊆ 𝑤))
254	253	reximdva 3276	. . . . 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 3184	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ∀𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})∃𝑣 ∈ ((nei‘𝐽)‘{𝑥})(((𝐽CnExt𝐾)‘𝐹) “ 𝑣) ⊆ 𝑤)
257	256	ralrimiva 3184	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐶 ∀𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})∃𝑣 ∈ ((nei‘𝐽)‘{𝑥})(((𝐽CnExt𝐾)‘𝐹) “ 𝑣) ⊆ 𝑤)
258	65, 42	cnnei 21892	. . 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 1367	. 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 259	1 ⊢ (𝜑 → ((𝐽CnExt𝐾)‘𝐹) ∈ (𝐽 Cn 𝐾))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ∀wral 3140 ∃wrex 3141 Vcvv 3496 ∩ cin 3937 ⊆ wss 3938 ∅c0 4293 {csn 4569 ∪ cuni 4840 class class class wbr 5068 dom cdm 5557 ran crn 5558 “ cima 5560 Fun wfun 6351 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 1_oc1o 8097 ≈ cen 8508 ↾_t crest 16696 fBascfbas 20535 filGencfg 20536 Topctop 21503 TopOnctopon 21520 clsccl 21628 neicnei 21707 Cn ccn 21834 Hauscha 21918 Regcreg 21919 Filcfil 22455 FilMap cfm 22543 fLim cflim 22544 fLimf cflf 22545 CnExtccnext 22669

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-1st 7691 df-2nd 7692 df-1o 8104 df-map 8410 df-pm 8411 df-en 8512 df-rest 16698 df-topgen 16719 df-fbas 20544 df-fg 20545 df-top 21504 df-topon 21521 df-cld 21629 df-ntr 21630 df-cls 21631 df-nei 21708 df-cn 21837 df-cnp 21838 df-haus 21925 df-reg 21926 df-fil 22456 df-fm 22548 df-flim 22549 df-flf 22550 df-cnext 22670

This theorem is referenced by: cnextucn 22914

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