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Theorem cnextcn 24205

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 778	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) → 𝜑)
2		simpll 778	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ (𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)}) ∧ 𝑑 ∈ ((nei‘𝐽)‘{𝑥}) ∧ ((cls‘𝐾)‘(𝐹 “ (𝑑 ∩ 𝐴))) ⊆ 𝑤)) → 𝜑)
3		simpr3 1215	. . . . . . . . 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 738	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ (𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)}) ∧ 𝑑 ∈ ((nei‘𝐽)‘{𝑥}) ∧ ((cls‘𝐾)‘(𝐹 “ (𝑑 ∩ 𝐴))) ⊆ 𝑤)) → 𝐽 ∈ Top)
6		simpr2 1214	. . . . . . . . . 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 23246	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝑑 ∈ ((nei‘𝐽)‘{𝑥})) → ∃𝑣 ∈ 𝐽 ({𝑥} ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑑))
8	5, 6, 7	syl2anc 595	. . . . . . . . 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 3459	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑥 ∈ V
10	9	snss 4751	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ 𝑣 ↔ {𝑥} ⊆ 𝑣)
11	10	biimpri 231	. . . . . . . . . . . . . . . . . 18 ⊢ ({𝑥} ⊆ 𝑣 → 𝑥 ∈ 𝑣)
12	11	anim1i 626	. . . . . . . . . . . . . . . . 17 ⊢ (({𝑥} ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑑) → (𝑥 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑑))
13	12	anim2i 628	. . . . . . . . . . . . . . . 16 ⊢ ((𝑣 ∈ 𝐽 ∧ ({𝑥} ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑑)) → (𝑣 ∈ 𝐽 ∧ (𝑥 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑑)))
14	13	anim2i 628	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐽 ∧ ({𝑥} ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑑))) → (𝜑 ∧ (𝑣 ∈ 𝐽 ∧ (𝑥 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑑))))
15	14	ex 417	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑣 ∈ 𝐽 ∧ ({𝑥} ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑑)) → (𝜑 ∧ (𝑣 ∈ 𝐽 ∧ (𝑥 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑑)))))
16		3anass 1111	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐽 ∧ 𝑥 ∈ 𝑣) ↔ (𝜑 ∧ (𝑣 ∈ 𝐽 ∧ 𝑥 ∈ 𝑣)))
17	16	anbi1i 635	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽 ∧ 𝑥 ∈ 𝑣) ∧ 𝑣 ⊆ 𝑑) ↔ ((𝜑 ∧ (𝑣 ∈ 𝐽 ∧ 𝑥 ∈ 𝑣)) ∧ 𝑣 ⊆ 𝑑))
18		anass 473	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐽 ∧ 𝑥 ∈ 𝑣)) ∧ 𝑣 ⊆ 𝑑) ↔ (𝜑 ∧ ((𝑣 ∈ 𝐽 ∧ 𝑥 ∈ 𝑣) ∧ 𝑣 ⊆ 𝑑)))
19		anass 473	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑣 ∈ 𝐽 ∧ 𝑥 ∈ 𝑣) ∧ 𝑣 ⊆ 𝑑) ↔ (𝑣 ∈ 𝐽 ∧ (𝑥 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑑)))
20	19	anbi2i 634	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑣 ∈ 𝐽 ∧ 𝑥 ∈ 𝑣) ∧ 𝑣 ⊆ 𝑑)) ↔ (𝜑 ∧ (𝑣 ∈ 𝐽 ∧ (𝑥 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑑))))
21	17, 18, 20	3bitri 300	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽 ∧ 𝑥 ∈ 𝑣) ∧ 𝑣 ⊆ 𝑑) ↔ (𝜑 ∧ (𝑣 ∈ 𝐽 ∧ (𝑥 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑑))))
22		opnneip 23257	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐽 ∈ Top ∧ 𝑣 ∈ 𝐽 ∧ 𝑥 ∈ 𝑣) → 𝑣 ∈ ((nei‘𝐽)‘{𝑥}))
23	4, 22	syl3an1 1181	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐽 ∧ 𝑥 ∈ 𝑣) → 𝑣 ∈ ((nei‘𝐽)‘{𝑥}))
24	23	adantr 485	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽 ∧ 𝑥 ∈ 𝑣) ∧ 𝑣 ⊆ 𝑑) → 𝑣 ∈ ((nei‘𝐽)‘{𝑥}))
25		simpr2 1214	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑣 ⊆ 𝑑 ∧ (𝜑 ∧ 𝑣 ∈ 𝐽 ∧ 𝑥 ∈ 𝑣)) → 𝑣 ∈ 𝐽)
26	25	ex 417	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 ⊆ 𝑑 → ((𝜑 ∧ 𝑣 ∈ 𝐽 ∧ 𝑥 ∈ 𝑣) → 𝑣 ∈ 𝐽))
27	26	imdistanri 579	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽 ∧ 𝑥 ∈ 𝑣) ∧ 𝑣 ⊆ 𝑑) → (𝑣 ∈ 𝐽 ∧ 𝑣 ⊆ 𝑑))
28	24, 27	jca 520	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽 ∧ 𝑥 ∈ 𝑣) ∧ 𝑣 ⊆ 𝑑) → (𝑣 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝑣 ∈ 𝐽 ∧ 𝑣 ⊆ 𝑑)))
29	21, 28	sylbir 238	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐽 ∧ (𝑥 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑑))) → (𝑣 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝑣 ∈ 𝐽 ∧ 𝑣 ⊆ 𝑑)))
30	15, 29	syl6 36	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑣 ∈ 𝐽 ∧ ({𝑥} ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑑)) → (𝑣 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝑣 ∈ 𝐽 ∧ 𝑣 ⊆ 𝑑))))
31	30	adantr 485	. . . . . . . . . . . 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 23469	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐾 ∈ Haus → 𝐾 ∈ Top)
34	32, 33	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐾 ∈ Top)
35		imassrn 6075	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐹 “ (𝑑 ∩ 𝐴)) ⊆ ran 𝐹
36		cnextf.5	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
37	36	frnd 6716	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ran 𝐹 ⊆ 𝐵)
38	35, 37	sstrid 3949	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐹 “ (𝑑 ∩ 𝐴)) ⊆ 𝐵)
39		ssrin 4195	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑣 ⊆ 𝑑 → (𝑣 ∩ 𝐴) ⊆ (𝑑 ∩ 𝐴))
40		imass2 6106	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑣 ∩ 𝐴) ⊆ (𝑑 ∩ 𝐴) → (𝐹 “ (𝑣 ∩ 𝐴)) ⊆ (𝐹 “ (𝑑 ∩ 𝐴)))
41	39, 40	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 ⊆ 𝑑 → (𝐹 “ (𝑣 ∩ 𝐴)) ⊆ (𝐹 “ (𝑑 ∩ 𝐴)))
42		cnextf.2	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐵 = ∪ 𝐾
43	42	clsss 23192	. . . . . . . . . . . . . . . . . 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 1449	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑣 ⊆ 𝑑) → ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))) ⊆ ((cls‘𝐾)‘(𝐹 “ (𝑑 ∩ 𝐴))))
45		sstr 3946	. . . . . . . . . . . . . . . . 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 591	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑣 ⊆ 𝑑) ∧ ((cls‘𝐾)‘(𝐹 “ (𝑑 ∩ 𝐴))) ⊆ 𝑤) → ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))) ⊆ 𝑤)
47	46	an32s 664	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ((cls‘𝐾)‘(𝐹 “ (𝑑 ∩ 𝐴))) ⊆ 𝑤) ∧ 𝑣 ⊆ 𝑑) → ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))) ⊆ 𝑤)
48	47	ex 417	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((cls‘𝐾)‘(𝐹 “ (𝑑 ∩ 𝐴))) ⊆ 𝑤) → (𝑣 ⊆ 𝑑 → ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))) ⊆ 𝑤))
49	48	anim2d 623	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((cls‘𝐾)‘(𝐹 “ (𝑑 ∩ 𝐴))) ⊆ 𝑤) → ((𝑣 ∈ 𝐽 ∧ 𝑣 ⊆ 𝑑) → (𝑣 ∈ 𝐽 ∧ ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))) ⊆ 𝑤)))
50	49	anim2d 623	. . . . . . . . . . . 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 48	. . . . . . . . . . 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 3175	. . . . . . . . . 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 411	. . . . . . . . 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 850	. . . . . . . 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 1381	. . . . . . 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 489	. . . . . . . . 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 794	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) ∧ 𝑢 ∈ (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴)) ∧ ((cls‘𝐾)‘(𝐹 “ 𝑢)) ⊆ 𝑤) → 𝜑)
58		simplr 780	. . . . . . . . 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 6060	. . . . . . . . . . . . . 14 ⊢ (𝑢 = (𝑑 ∩ 𝐴) → (𝐹 “ 𝑢) = (𝐹 “ (𝑑 ∩ 𝐴)))
60	59	fveq2d 6887	. . . . . . . . . . . . 13 ⊢ (𝑢 = (𝑑 ∩ 𝐴) → ((cls‘𝐾)‘(𝐹 “ 𝑢)) = ((cls‘𝐾)‘(𝐹 “ (𝑑 ∩ 𝐴))))
61	60	sseq1d 3969	. . . . . . . . . . . 12 ⊢ (𝑢 = (𝑑 ∩ 𝐴) → (((cls‘𝐾)‘(𝐹 “ 𝑢)) ⊆ 𝑤 ↔ ((cls‘𝐾)‘(𝐹 “ (𝑑 ∩ 𝐴))) ⊆ 𝑤))
62	61	biimpcd 252	. . . . . . . . . . 11 ⊢ (((cls‘𝐾)‘(𝐹 “ 𝑢)) ⊆ 𝑤 → (𝑢 = (𝑑 ∩ 𝐴) → ((cls‘𝐾)‘(𝐹 “ (𝑑 ∩ 𝐴))) ⊆ 𝑤))
63	62	reximdv 3180	. . . . . . . . . 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 6898	. . . . . . . . . . . 12 ⊢ (𝜑 → ((nei‘𝐽)‘{𝑥}) ∈ V)
65		cnextf.1	. . . . . . . . . . . . . . . 16 ⊢ 𝐶 = ∪ 𝐽
66	65	toptopon 23055	. . . . . . . . . . . . . . 15 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝐶))
67	4, 66	sylib 221	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝐶))
68	67	elfvexd 6919	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐶 ∈ V)
69		cnextf.a	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 ⊆ 𝐶)
70	68, 69	ssexd 5296	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ V)
71		elrest 17481	. . . . . . . . . . . 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 595	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑢 ∈ (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴) ↔ ∃𝑑 ∈ ((nei‘𝐽)‘{𝑥})𝑢 = (𝑑 ∩ 𝐴)))
73	72	biimpa 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑢 ∈ (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴)) → ∃𝑑 ∈ ((nei‘𝐽)‘{𝑥})𝑢 = (𝑑 ∩ 𝐴))
74	63, 73	impel 514	. . . . . . . . 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 849	. . . . . . . 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 2846	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐶 ↔ 𝑦 ∈ 𝐶))
78	77	anbi2d 641	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑦 → ((𝜑 ∧ 𝑥 ∈ 𝐶) ↔ (𝜑 ∧ 𝑦 ∈ 𝐶)))
79		sneq 4600	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦})
80	79	fveq2d 6887	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑦 → ((nei‘𝐽)‘{𝑥}) = ((nei‘𝐽)‘{𝑦}))
81	80	oveq1d 7427	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = 𝑦 → (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴) = (((nei‘𝐽)‘{𝑦}) ↾_t 𝐴))
82	81	oveq2d 7428	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑦 → (𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴)) = (𝐾 fLimf (((nei‘𝐽)‘{𝑦}) ↾_t 𝐴)))
83	82	fveq1d 6885	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑦 → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴))‘𝐹) = ((𝐾 fLimf (((nei‘𝐽)‘{𝑦}) ↾_t 𝐴))‘𝐹))
84	83	neeq1d 3017	. . . . . . . . . . . . . . . . . 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 2019	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑦}) ↾_t 𝐴))‘𝐹) ≠ ∅)
88	65, 42, 4, 32, 36, 69, 76, 87	cnextfvval 24203	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (((𝐽CnExt𝐾)‘𝐹)‘𝑥) = ∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴))‘𝐹))
89		fvex 6896	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴))‘𝐹) ∈ V
90	89	uniex 7741	. . . . . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐾 ∈ Haus)
93	76	eleq2d 2849	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↔ 𝑥 ∈ 𝐶))
94	93	biimpar 482	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝑥 ∈ ((cls‘𝐽)‘𝐴))
95	67	adantr 485	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐽 ∈ (TopOn‘𝐶))
96	69	adantr 485	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐴 ⊆ 𝐶)
97		simpr 489	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝑥 ∈ 𝐶)
98		trnei 24030	. . . . . . . . . . . . . . . . . . . 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 1398	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↔ (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴) ∈ (Fil‘𝐴)))
100	94, 99	mpbid 235	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴) ∈ (Fil‘𝐴))
101	36	adantr 485	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐹:𝐴⟶𝐵)
102	42	hausflf2 24136	. . . . . . . . . . . . . . . . . 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 1400	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴))‘𝐹) ≈ 1_o)
104		en1b 9023	. . . . . . . . . . . . . . . . 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 221	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴))‘𝐹) = {∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴))‘𝐹)})
106	91, 105	eleqtrrid 2870	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴))‘𝐹) ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴))‘𝐹))
107	88, 106	eqeltrd 2863	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴))‘𝐹))
108	42	toptopon 23055	. . . . . . . . . . . . . . . . 17 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝐵))
109	34, 108	sylib 221	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝐵))
110	109	adantr 485	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐾 ∈ (TopOn‘𝐵))
111		flfnei 24129	. . . . . . . . . . . . . . 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 1398	. . . . . . . . . . . . . 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 235	. . . . . . . . . . . . 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 500	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ∀𝑏 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})∃𝑢 ∈ (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴)(𝐹 “ 𝑢) ⊆ 𝑏)
115	114	r19.21bi 3257	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑏 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) → ∃𝑢 ∈ (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴)(𝐹 “ 𝑢) ⊆ 𝑏)
116	115	ad4ant13 763	. . . . . . . . . 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 742	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑏 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑤) → 𝐾 ∈ Top)
118		simplr 780	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑏 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑤) → 𝑏 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)}))
119	42	neii1 23244	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ Top ∧ 𝑏 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) → 𝑏 ⊆ 𝐵)
120	117, 118, 119	syl2anc 595	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑏 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑤) → 𝑏 ⊆ 𝐵)
121		simpr 489	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑏 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑤) → ((cls‘𝐾)‘𝑏) ⊆ 𝑤)
122	42	clsss 23192	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ Top ∧ 𝑏 ⊆ 𝐵 ∧ (𝐹 “ 𝑢) ⊆ 𝑏) → ((cls‘𝐾)‘(𝐹 “ 𝑢)) ⊆ ((cls‘𝐾)‘𝑏))
123		sstr 3946	. . . . . . . . . . . . . . . 16 ⊢ ((((cls‘𝐾)‘(𝐹 “ 𝑢)) ⊆ ((cls‘𝐾)‘𝑏) ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑤) → ((cls‘𝐾)‘(𝐹 “ 𝑢)) ⊆ 𝑤)
124	122, 123	sylan 591	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ Top ∧ 𝑏 ⊆ 𝐵 ∧ (𝐹 “ 𝑢) ⊆ 𝑏) ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑤) → ((cls‘𝐾)‘(𝐹 “ 𝑢)) ⊆ 𝑤)
125	124	3an1rs 1378	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ Top ∧ 𝑏 ⊆ 𝐵 ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑤) ∧ (𝐹 “ 𝑢) ⊆ 𝑏) → ((cls‘𝐾)‘(𝐹 “ 𝑢)) ⊆ 𝑤)
126	125	ex 417	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ Top ∧ 𝑏 ⊆ 𝐵 ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑤) → ((𝐹 “ 𝑢) ⊆ 𝑏 → ((cls‘𝐾)‘(𝐹 “ 𝑢)) ⊆ 𝑤))
127	126	reximdv 3180	. . . . . . . . . . . 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 1398	. . . . . . . . . . 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 731	. . . . . . . . . 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 16	. . . . . . . . 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 738	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) → 𝐾 ∈ Top)
132		cnextcn.8	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐾 ∈ Reg)
133	132	ad2antrr 738	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) → 𝐾 ∈ Reg)
134	133	ad2antrr 738	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) ∧ 𝑐 ∈ 𝐾) ∧ ((((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑐 ∧ 𝑐 ⊆ 𝑤)) → 𝐾 ∈ Reg)
135		simplr 780	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) ∧ 𝑐 ∈ 𝐾) ∧ ((((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑐 ∧ 𝑐 ⊆ 𝑤)) → 𝑐 ∈ 𝐾)
136		simprl 782	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) ∧ 𝑐 ∈ 𝐾) ∧ ((((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑐 ∧ 𝑐 ⊆ 𝑤)) → (((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑐)
137		regsep 23472	. . . . . . . . . . . . 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 1398	. . . . . . . . . . . 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 3946	. . . . . . . . . . . . . . . 16 ⊢ ((((cls‘𝐾)‘𝑏) ⊆ 𝑐 ∧ 𝑐 ⊆ 𝑤) → ((cls‘𝐾)‘𝑏) ⊆ 𝑤)
140	139	expcom 418	. . . . . . . . . . . . . . 15 ⊢ (𝑐 ⊆ 𝑤 → (((cls‘𝐾)‘𝑏) ⊆ 𝑐 → ((cls‘𝐾)‘𝑏) ⊆ 𝑤))
141	140	anim2d 623	. . . . . . . . . . . . . 14 ⊢ (𝑐 ⊆ 𝑤 → (((((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑏 ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑐) → ((((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑏 ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑤)))
142	141	reximdv 3180	. . . . . . . . . . . . 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 741	. . . . . . . . . . . 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 16	. . . . . . . . . . 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 489	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) → 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)}))
146		neii2 23246	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ Top ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) → ∃𝑐 ∈ 𝐾 ({(((𝐽CnExt𝐾)‘𝐹)‘𝑥)} ⊆ 𝑐 ∧ 𝑐 ⊆ 𝑤))
147		fvex 6896	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ V
148	147	snss 4751	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑐 ↔ {(((𝐽CnExt𝐾)‘𝐹)‘𝑥)} ⊆ 𝑐)
149	148	anbi1i 635	. . . . . . . . . . . . . . 15 ⊢ (((((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑐 ∧ 𝑐 ⊆ 𝑤) ↔ ({(((𝐽CnExt𝐾)‘𝐹)‘𝑥)} ⊆ 𝑐 ∧ 𝑐 ⊆ 𝑤))
150	149	biimpri 231	. . . . . . . . . . . . . 14 ⊢ (({(((𝐽CnExt𝐾)‘𝐹)‘𝑥)} ⊆ 𝑐 ∧ 𝑐 ⊆ 𝑤) → ((((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑐 ∧ 𝑐 ⊆ 𝑤))
151	150	reximi 3103	. . . . . . . . . . . . 13 ⊢ (∃𝑐 ∈ 𝐾 ({(((𝐽CnExt𝐾)‘𝐹)‘𝑥)} ⊆ 𝑐 ∧ 𝑐 ⊆ 𝑤) → ∃𝑐 ∈ 𝐾 ((((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑐 ∧ 𝑐 ⊆ 𝑤))
152	146, 151	syl 18	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ Top ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) → ∃𝑐 ∈ 𝐾 ((((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑐 ∧ 𝑐 ⊆ 𝑤))
153	131, 145, 152	syl2anc 595	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) → ∃𝑐 ∈ 𝐾 ((((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑐 ∧ 𝑐 ⊆ 𝑤))
154	144, 153	r19.29a 3173	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) → ∃𝑏 ∈ 𝐾 ((((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑏 ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑤))
155		anass 473	. . . . . . . . . . . 12 ⊢ (((𝑏 ∈ 𝐾 ∧ (((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑏) ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑤) ↔ (𝑏 ∈ 𝐾 ∧ ((((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑏 ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑤)))
156		opnneip 23257	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ Top ∧ 𝑏 ∈ 𝐾 ∧ (((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑏) → 𝑏 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)}))
157	156	3expib 1140	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ Top → ((𝑏 ∈ 𝐾 ∧ (((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑏) → 𝑏 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})))
158	157	anim1d 622	. . . . . . . . . . . 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 246	. . . . . . . . . . 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 3175	. . . . . . . . . 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 66	. . . . . . . . 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 3173	. . . . . . . 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 3173	. . . . . . 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 3173	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) → ∃𝑣 ∈ ((nei‘𝐽)‘{𝑥})(𝑣 ∈ 𝐽 ∧ ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))) ⊆ 𝑤))
165		simplr 780	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))) ⊆ 𝑤) ∧ 𝑧 ∈ 𝑣) → ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))) ⊆ 𝑤)
166		simpll 778	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ 𝑧 ∈ 𝑣) → 𝜑)
167	4	ad2antrr 738	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ 𝑧 ∈ 𝑣) → 𝐽 ∈ Top)
168		simplr 780	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ 𝑧 ∈ 𝑣) → 𝑣 ∈ 𝐽)
169	65	eltopss 23045	. . . . . . . . . . . . . . 15 ⊢ ((𝐽 ∈ Top ∧ 𝑣 ∈ 𝐽) → 𝑣 ⊆ 𝐶)
170	167, 168, 169	syl2anc 595	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ 𝑧 ∈ 𝑣) → 𝑣 ⊆ 𝐶)
171		simpr 489	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ 𝑧 ∈ 𝑣) → 𝑧 ∈ 𝑣)
172	170, 171	sseldd 3939	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ 𝑧 ∈ 𝑣) → 𝑧 ∈ 𝐶)
173		fvexd 6898	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ 𝑧 ∈ 𝑣) → ((nei‘𝐽)‘{𝑧}) ∈ V)
174	70	ad2antrr 738	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ 𝑧 ∈ 𝑣) → 𝐴 ∈ V)
175		opnneip 23257	. . . . . . . . . . . . . . . 16 ⊢ ((𝐽 ∈ Top ∧ 𝑣 ∈ 𝐽 ∧ 𝑧 ∈ 𝑣) → 𝑣 ∈ ((nei‘𝐽)‘{𝑧}))
176	4, 175	syl3an1 1181	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐽 ∧ 𝑧 ∈ 𝑣) → 𝑣 ∈ ((nei‘𝐽)‘{𝑧}))
177	176	3expa 1136	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ 𝑧 ∈ 𝑣) → 𝑣 ∈ ((nei‘𝐽)‘{𝑧}))
178		elrestr 17482	. . . . . . . . . . . . . 14 ⊢ ((((nei‘𝐽)‘{𝑧}) ∈ V ∧ 𝐴 ∈ V ∧ 𝑣 ∈ ((nei‘𝐽)‘{𝑧})) → (𝑣 ∩ 𝐴) ∈ (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴))
179	173, 174, 177, 178	syl3anc 1398	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ 𝑧 ∈ 𝑣) → (𝑣 ∩ 𝐴) ∈ (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴))
180	65, 42, 4, 32, 36, 69, 76, 86	cnextfvval 24203	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → (((𝐽CnExt𝐾)‘𝐹)‘𝑧) = ∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴))‘𝐹))
181	180	adantr 485	. . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → 𝐾 ∈ Haus)
183	76	eleq2d 2849	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝑧 ∈ ((cls‘𝐽)‘𝐴) ↔ 𝑧 ∈ 𝐶))
184	183	biimpar 482	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → 𝑧 ∈ ((cls‘𝐽)‘𝐴))
185	67	adantr 485	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → 𝐽 ∈ (TopOn‘𝐶))
186	69	adantr 485	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → 𝐴 ⊆ 𝐶)
187		simpr 489	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → 𝑧 ∈ 𝐶)
188		trnei 24030	. . . . . . . . . . . . . . . . . . . . 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 1398	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → (𝑧 ∈ ((cls‘𝐽)‘𝐴) ↔ (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴) ∈ (Fil‘𝐴)))
190	184, 189	mpbid 235	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴) ∈ (Fil‘𝐴))
191	36	adantr 485	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → 𝐹:𝐴⟶𝐵)
192		eleq1w 2846	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐶 ↔ 𝑧 ∈ 𝐶))
193	192	anbi2d 641	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = 𝑧 → ((𝜑 ∧ 𝑥 ∈ 𝐶) ↔ (𝜑 ∧ 𝑧 ∈ 𝐶)))
194		sneq 4600	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑥 = 𝑧 → {𝑥} = {𝑧})
195	194	fveq2d 6887	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥 = 𝑧 → ((nei‘𝐽)‘{𝑥}) = ((nei‘𝐽)‘{𝑧}))
196	195	oveq1d 7427	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 = 𝑧 → (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴) = (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴))
197	196	oveq2d 7428	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = 𝑧 → (𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴)) = (𝐾 fLimf (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴)))
198	197	fveq1d 6885	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑧 → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴))‘𝐹) = ((𝐾 fLimf (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴))‘𝐹))
199	198	neeq1d 3017	. . . . . . . . . . . . . . . . . . . . 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 2019	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴))‘𝐹) ≠ ∅)
202	42	hausflf2 24136	. . . . . . . . . . . . . . . . . . 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 1400	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴))‘𝐹) ≈ 1_o)
204		en1b 9023	. . . . . . . . . . . . . . . . . 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 221	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴))‘𝐹) = {∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴))‘𝐹)})
206	205	adantr 485	. . . . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → 𝐾 ∈ (TopOn‘𝐵))
208		flfval 24128	. . . . . . . . . . . . . . . . . . 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 1398	. . . . . . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . . . . 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 7742	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ∪ 𝐾 ∈ V)
212	211	ad2antrr 738	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐶) ∧ (𝑣 ∩ 𝐴) ∈ (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴)) → ∪ 𝐾 ∈ V)
213	42, 212	eqeltrid 2867	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐶) ∧ (𝑣 ∩ 𝐴) ∈ (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴)) → 𝐵 ∈ V)
214	190	adantr 485	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐶) ∧ (𝑣 ∩ 𝐴) ∈ (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴)) → (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴) ∈ (Fil‘𝐴))
215		filfbas 23986	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((nei‘𝐽)‘{𝑧}) ↾_t 𝐴) ∈ (Fil‘𝐴) → (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴) ∈ (fBas‘𝐴))
216	214, 215	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐶) ∧ (𝑣 ∩ 𝐴) ∈ (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴)) → (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴) ∈ (fBas‘𝐴))
217	36	ad2antrr 738	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐶) ∧ (𝑣 ∩ 𝐴) ∈ (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴)) → 𝐹:𝐴⟶𝐵)
218		simpr 489	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐶) ∧ (𝑣 ∩ 𝐴) ∈ (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴)) → (𝑣 ∩ 𝐴) ∈ (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴))
219		fgfil 24013	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((nei‘𝐽)‘{𝑧}) ↾_t 𝐴) ∈ (Fil‘𝐴) → (𝐴filGen(((nei‘𝐽)‘{𝑧}) ↾_t 𝐴)) = (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴))
220	190, 219	syl 18	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → (𝐴filGen(((nei‘𝐽)‘{𝑧}) ↾_t 𝐴)) = (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴))
221	220	adantr 485	. . . . . . . . . . . . . . . . . . . 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 2866	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐶) ∧ (𝑣 ∩ 𝐴) ∈ (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴)) → (𝑣 ∩ 𝐴) ∈ (𝐴filGen(((nei‘𝐽)‘{𝑧}) ↾_t 𝐴)))
223		eqid 2763	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐴filGen(((nei‘𝐽)‘{𝑧}) ↾_t 𝐴)) = (𝐴filGen(((nei‘𝐽)‘{𝑧}) ↾_t 𝐴))
224	223	imaelfm 24089	. . . . . . . . . . . . . . . . . . 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 1400	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐶) ∧ (𝑣 ∩ 𝐴) ∈ (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴)) → (𝐹 “ (𝑣 ∩ 𝐴)) ∈ ((𝐵 FilMap 𝐹)‘(((nei‘𝐽)‘{𝑧}) ↾_t 𝐴)))
226		flimclsi 24116	. . . . . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . . . 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 3972	. . . . . . . . . . . . . . . 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 3973	. . . . . . . . . . . . . . 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 6896	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐾 fLimf (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴))‘𝐹) ∈ V
231	230	uniex 7741	. . . . . . . . . . . . . . . 16 ⊢ ∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴))‘𝐹) ∈ V
232	231	snss 4751	. . . . . . . . . . . . . . 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 237	. . . . . . . . . . . . . 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 2863	. . . . . . . . . . . . 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 850	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ 𝑧 ∈ 𝑣) → (((𝐽CnExt𝐾)‘𝐹)‘𝑧) ∈ ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))))
236	235	adantlr 727	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))) ⊆ 𝑤) ∧ 𝑧 ∈ 𝑣) → (((𝐽CnExt𝐾)‘𝐹)‘𝑧) ∈ ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))))
237	165, 236	sseldd 3939	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))) ⊆ 𝑤) ∧ 𝑧 ∈ 𝑣) → (((𝐽CnExt𝐾)‘𝐹)‘𝑧) ∈ 𝑤)
238	237	ralrimiva 3157	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))) ⊆ 𝑤) → ∀𝑧 ∈ 𝑣 (((𝐽CnExt𝐾)‘𝐹)‘𝑧) ∈ 𝑤)
239	238	expl 462	. . . . . . . 8 ⊢ (𝜑 → ((𝑣 ∈ 𝐽 ∧ ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))) ⊆ 𝑤) → ∀𝑧 ∈ 𝑣 (((𝐽CnExt𝐾)‘𝐹)‘𝑧) ∈ 𝑤))
240	239	reximdv 3180	. . . . . . 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 738	. . . . . 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 16	. . . . 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 24204	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐽CnExt𝐾)‘𝐹):𝐶⟶𝐵)
244	243	ffund 6712	. . . . . . . . 9 ⊢ (𝜑 → Fun ((𝐽CnExt𝐾)‘𝐹))
245	244	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ ((nei‘𝐽)‘{𝑥})) → Fun ((𝐽CnExt𝐾)‘𝐹))
246	65	neii1 23244	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝑣 ∈ ((nei‘𝐽)‘{𝑥})) → 𝑣 ⊆ 𝐶)
247	4, 246	sylan 591	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ ((nei‘𝐽)‘{𝑥})) → 𝑣 ⊆ 𝐶)
248	243	fdmd 6718	. . . . . . . . . 10 ⊢ (𝜑 → dom ((𝐽CnExt𝐾)‘𝐹) = 𝐶)
249	248	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ ((nei‘𝐽)‘{𝑥})) → dom ((𝐽CnExt𝐾)‘𝐹) = 𝐶)
250	247, 249	sseqtrrd 3975	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ ((nei‘𝐽)‘{𝑥})) → 𝑣 ⊆ dom ((𝐽CnExt𝐾)‘𝐹))
251		funimass4 6947	. . . . . . . 8 ⊢ ((Fun ((𝐽CnExt𝐾)‘𝐹) ∧ 𝑣 ⊆ dom ((𝐽CnExt𝐾)‘𝐹)) → ((((𝐽CnExt𝐾)‘𝐹) “ 𝑣) ⊆ 𝑤 ↔ ∀𝑧 ∈ 𝑣 (((𝐽CnExt𝐾)‘𝐹)‘𝑧) ∈ 𝑤))
252	245, 250, 251	syl2anc 595	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ ((nei‘𝐽)‘{𝑥})) → ((((𝐽CnExt𝐾)‘𝐹) “ 𝑣) ⊆ 𝑤 ↔ ∀𝑧 ∈ 𝑣 (((𝐽CnExt𝐾)‘𝐹)‘𝑧) ∈ 𝑤))
253	252	biimprd 251	. . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ ((nei‘𝐽)‘{𝑥})) → (∀𝑧 ∈ 𝑣 (((𝐽CnExt𝐾)‘𝐹)‘𝑧) ∈ 𝑤 → (((𝐽CnExt𝐾)‘𝐹) “ 𝑣) ⊆ 𝑤))
254	253	reximdva 3178	. . . . 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 66	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) → ∃𝑣 ∈ ((nei‘𝐽)‘{𝑥})(((𝐽CnExt𝐾)‘𝐹) “ 𝑣) ⊆ 𝑤)
256	255	ralrimiva 3157	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ∀𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})∃𝑣 ∈ ((nei‘𝐽)‘{𝑥})(((𝐽CnExt𝐾)‘𝐹) “ 𝑣) ⊆ 𝑤)
257	256	ralrimiva 3157	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐶 ∀𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})∃𝑣 ∈ ((nei‘𝐽)‘{𝑥})(((𝐽CnExt𝐾)‘𝐹) “ 𝑣) ⊆ 𝑤)
258	65, 42	cnnei 23420	. . 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 1398	. 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 260	1 ⊢ (𝜑 → ((𝐽CnExt𝐾)‘𝐹) ∈ (𝐽 Cn 𝐾))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 ∧ w3a 1103 = wceq 1570 ∈ wcel 2143 ≠ wne 2958 ∀wral 3079 ∃wrex 3089 Vcvv 3455 ∩ cin 3905 ⊆ wss 3906 ∅c0 4287 {csn 4590 ∪ cuni 4873 class class class wbr 5110 dom cdm 5663 ran crn 5664 “ cima 5666 Fun wfun 6532 ⟶wf 6534 ‘cfv 6538 (class class class)co 7412 1_oc1o 8447 ≈ cen 8941 ↾_t crest 17474 fBascfbas 21491 filGencfg 21492 Topctop 23031 TopOnctopon 23048 clsccl 23156 neicnei 23235 Cn ccn 23362 Hauscha 23446 Regcreg 23447 Filcfil 23983 FilMap cfm 24071 fLim cflim 24072 fLimf cflf 24073 CnExtccnext 24197

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5239 ax-sep 5258 ax-nul 5270 ax-pow 5338 ax-pr 5406 ax-un 7734

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3746 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-nul 4288 df-if 4489 df-pw 4565 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4874 df-int 4914 df-iun 4959 df-iin 4960 df-br 5111 df-opab 5175 df-mpt 5194 df-id 5558 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-ov 7415 df-oprab 7416 df-mpo 7417 df-1st 7987 df-2nd 7988 df-1o 8454 df-map 8827 df-pm 8828 df-en 8945 df-rest 17476 df-topgen 17497 df-fbas 21500 df-fg 21501 df-top 23032 df-topon 23049 df-cld 23157 df-ntr 23158 df-cls 23159 df-nei 23236 df-cn 23365 df-cnp 23366 df-haus 23453 df-reg 23454 df-fil 23984 df-fm 24076 df-flim 24077 df-flf 24078 df-cnext 24198

This theorem is referenced by: cnextucn 24440

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