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Theorem cnextcn 24032

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 767	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) → 𝜑)
2		simpll 767	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ (𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)}) ∧ 𝑑 ∈ ((nei‘𝐽)‘{𝑥}) ∧ ((cls‘𝐾)‘(𝐹 “ (𝑑 ∩ 𝐴))) ⊆ 𝑤)) → 𝜑)
3		simpr3 1198	. . . . . . . . 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 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ (𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)}) ∧ 𝑑 ∈ ((nei‘𝐽)‘{𝑥}) ∧ ((cls‘𝐾)‘(𝐹 “ (𝑑 ∩ 𝐴))) ⊆ 𝑤)) → 𝐽 ∈ Top)
6		simpr2 1197	. . . . . . . . . 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 23073	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝑑 ∈ ((nei‘𝐽)‘{𝑥})) → ∃𝑣 ∈ 𝐽 ({𝑥} ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑑))
8	5, 6, 7	syl2anc 585	. . . . . . . . 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 3433	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑥 ∈ V
10	9	snss 4728	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ 𝑣 ↔ {𝑥} ⊆ 𝑣)
11	10	biimpri 228	. . . . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑣 ∈ 𝐽 ∧ ({𝑥} ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑑)) → (𝜑 ∧ (𝑣 ∈ 𝐽 ∧ (𝑥 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑑)))))
16		3anass 1095	. . . . . . . . . . . . . . . . 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 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 624	. . . . . . . . . . . . . . . 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 23084	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐽 ∈ Top ∧ 𝑣 ∈ 𝐽 ∧ 𝑥 ∈ 𝑣) → 𝑣 ∈ ((nei‘𝐽)‘{𝑥}))
23	4, 22	syl3an1 1164	. . . . . . . . . . . . . . . . 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 1197	. . . . . . . . . . . . . . . . . 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 23296	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐾 ∈ Haus → 𝐾 ∈ Top)
34	32, 33	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐾 ∈ Top)
35		imassrn 6036	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐹 “ (𝑑 ∩ 𝐴)) ⊆ ran 𝐹
36		cnextf.5	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
37	36	frnd 6676	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ran 𝐹 ⊆ 𝐵)
38	35, 37	sstrid 3933	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐹 “ (𝑑 ∩ 𝐴)) ⊆ 𝐵)
39		ssrin 4182	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑣 ⊆ 𝑑 → (𝑣 ∩ 𝐴) ⊆ (𝑑 ∩ 𝐴))
40		imass2 6067	. . . . . . . . . . . . . . . . . . 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 23019	. . . . . . . . . . . . . . . . . 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 1425	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑣 ⊆ 𝑑) → ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))) ⊆ ((cls‘𝐾)‘(𝐹 “ (𝑑 ∩ 𝐴))))
45		sstr 3930	. . . . . . . . . . . . . . . . 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 581	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑣 ⊆ 𝑑) ∧ ((cls‘𝐾)‘(𝐹 “ (𝑑 ∩ 𝐴))) ⊆ 𝑤) → ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))) ⊆ 𝑤)
47	46	an32s 653	. . . . . . . . . . . . . . 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 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 3147	. . . . . . . . . 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 838	. . . . . . . 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 1362	. . . . . . 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 783	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) ∧ 𝑢 ∈ (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴)) ∧ ((cls‘𝐾)‘(𝐹 “ 𝑢)) ⊆ 𝑤) → 𝜑)
58		simplr 769	. . . . . . . . 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 6021	. . . . . . . . . . . . . 14 ⊢ (𝑢 = (𝑑 ∩ 𝐴) → (𝐹 “ 𝑢) = (𝐹 “ (𝑑 ∩ 𝐴)))
60	59	fveq2d 6844	. . . . . . . . . . . . 13 ⊢ (𝑢 = (𝑑 ∩ 𝐴) → ((cls‘𝐾)‘(𝐹 “ 𝑢)) = ((cls‘𝐾)‘(𝐹 “ (𝑑 ∩ 𝐴))))
61	60	sseq1d 3953	. . . . . . . . . . . 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 3152	. . . . . . . . . 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 6855	. . . . . . . . . . . 12 ⊢ (𝜑 → ((nei‘𝐽)‘{𝑥}) ∈ V)
65		cnextf.1	. . . . . . . . . . . . . . . 16 ⊢ 𝐶 = ∪ 𝐽
66	65	toptopon 22882	. . . . . . . . . . . . . . 15 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝐶))
67	4, 66	sylib 218	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝐶))
68	67	elfvexd 6876	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐶 ∈ V)
69		cnextf.a	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 ⊆ 𝐶)
70	68, 69	ssexd 5265	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ V)
71		elrest 17390	. . . . . . . . . . . 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 585	. . . . . . . . . . 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 837	. . . . . . . 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 2819	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐶 ↔ 𝑦 ∈ 𝐶))
78	77	anbi2d 631	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑦 → ((𝜑 ∧ 𝑥 ∈ 𝐶) ↔ (𝜑 ∧ 𝑦 ∈ 𝐶)))
79		sneq 4577	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦})
80	79	fveq2d 6844	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑦 → ((nei‘𝐽)‘{𝑥}) = ((nei‘𝐽)‘{𝑦}))
81	80	oveq1d 7382	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = 𝑦 → (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴) = (((nei‘𝐽)‘{𝑦}) ↾_t 𝐴))
82	81	oveq2d 7383	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑦 → (𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴)) = (𝐾 fLimf (((nei‘𝐽)‘{𝑦}) ↾_t 𝐴)))
83	82	fveq1d 6842	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑦 → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴))‘𝐹) = ((𝐾 fLimf (((nei‘𝐽)‘{𝑦}) ↾_t 𝐴))‘𝐹))
84	83	neeq1d 2991	. . . . . . . . . . . . . . . . . 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 1991	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑦}) ↾_t 𝐴))‘𝐹) ≠ ∅)
88	65, 42, 4, 32, 36, 69, 76, 87	cnextfvval 24030	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (((𝐽CnExt𝐾)‘𝐹)‘𝑥) = ∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴))‘𝐹))
89		fvex 6853	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴))‘𝐹) ∈ V
90	89	uniex 7695	. . . . . . . . . . . . . . . . 17 ⊢ ∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴))‘𝐹) ∈ V
91	90	snid 4606	. . . . . . . . . . . . . . . 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 2822	. . . . . . . . . . . . . . . . . . . 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 23857	. . . . . . . . . . . . . . . . . . . 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 1374	. . . . . . . . . . . . . . . . . . 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 23963	. . . . . . . . . . . . . . . . . 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 1376	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴))‘𝐹) ≈ 1_o)
104		en1b 8972	. . . . . . . . . . . . . . . . 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 2843	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴))‘𝐹) ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴))‘𝐹))
107	88, 106	eqeltrd 2836	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴))‘𝐹))
108	42	toptopon 22882	. . . . . . . . . . . . . . . . 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 23956	. . . . . . . . . . . . . . 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 1374	. . . . . . . . . . . . . 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 3229	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑏 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) → ∃𝑢 ∈ (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴)(𝐹 “ 𝑢) ⊆ 𝑏)
116	115	ad4ant13 752	. . . . . . . . . 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 731	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑏 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑤) → 𝐾 ∈ Top)
118		simplr 769	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑏 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑤) → 𝑏 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)}))
119	42	neii1 23071	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ Top ∧ 𝑏 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) → 𝑏 ⊆ 𝐵)
120	117, 118, 119	syl2anc 585	. . . . . . . . . . . 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 23019	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ Top ∧ 𝑏 ⊆ 𝐵 ∧ (𝐹 “ 𝑢) ⊆ 𝑏) → ((cls‘𝐾)‘(𝐹 “ 𝑢)) ⊆ ((cls‘𝐾)‘𝑏))
123		sstr 3930	. . . . . . . . . . . . . . . 16 ⊢ ((((cls‘𝐾)‘(𝐹 “ 𝑢)) ⊆ ((cls‘𝐾)‘𝑏) ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑤) → ((cls‘𝐾)‘(𝐹 “ 𝑢)) ⊆ 𝑤)
124	122, 123	sylan 581	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ Top ∧ 𝑏 ⊆ 𝐵 ∧ (𝐹 “ 𝑢) ⊆ 𝑏) ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑤) → ((cls‘𝐾)‘(𝐹 “ 𝑢)) ⊆ 𝑤)
125	124	3an1rs 1361	. . . . . . . . . . . . . 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 3152	. . . . . . . . . . . 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 1374	. . . . . . . . . . 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 720	. . . . . . . . . 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 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) → 𝐾 ∈ Top)
132		cnextcn.8	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐾 ∈ Reg)
133	132	ad2antrr 727	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) → 𝐾 ∈ Reg)
134	133	ad2antrr 727	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) ∧ 𝑐 ∈ 𝐾) ∧ ((((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑐 ∧ 𝑐 ⊆ 𝑤)) → 𝐾 ∈ Reg)
135		simplr 769	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) ∧ 𝑐 ∈ 𝐾) ∧ ((((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑐 ∧ 𝑐 ⊆ 𝑤)) → 𝑐 ∈ 𝐾)
136		simprl 771	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) ∧ 𝑐 ∈ 𝐾) ∧ ((((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑐 ∧ 𝑐 ⊆ 𝑤)) → (((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑐)
137		regsep 23299	. . . . . . . . . . . . 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 1374	. . . . . . . . . . . 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 3930	. . . . . . . . . . . . . . . 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 613	. . . . . . . . . . . . . 14 ⊢ (𝑐 ⊆ 𝑤 → (((((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑏 ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑐) → ((((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑏 ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑤)))
142	141	reximdv 3152	. . . . . . . . . . . . 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 730	. . . . . . . . . . . 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 23073	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ Top ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) → ∃𝑐 ∈ 𝐾 ({(((𝐽CnExt𝐾)‘𝐹)‘𝑥)} ⊆ 𝑐 ∧ 𝑐 ⊆ 𝑤))
147		fvex 6853	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ V
148	147	snss 4728	. . . . . . . . . . . . . . . 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 228	. . . . . . . . . . . . . 14 ⊢ (({(((𝐽CnExt𝐾)‘𝐹)‘𝑥)} ⊆ 𝑐 ∧ 𝑐 ⊆ 𝑤) → ((((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑐 ∧ 𝑐 ⊆ 𝑤))
151	150	reximi 3075	. . . . . . . . . . . . 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 585	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) → ∃𝑐 ∈ 𝐾 ((((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑐 ∧ 𝑐 ⊆ 𝑤))
154	144, 153	r19.29a 3145	. . . . . . . . . 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 23084	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ Top ∧ 𝑏 ∈ 𝐾 ∧ (((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑏) → 𝑏 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)}))
157	156	3expib 1123	. . . . . . . . . . . . 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	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 3147	. . . . . . . . . 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 3145	. . . . . . . 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 3145	. . . . . . 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 3145	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) → ∃𝑣 ∈ ((nei‘𝐽)‘{𝑥})(𝑣 ∈ 𝐽 ∧ ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))) ⊆ 𝑤))
165		simplr 769	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))) ⊆ 𝑤) ∧ 𝑧 ∈ 𝑣) → ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))) ⊆ 𝑤)
166		simpll 767	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ 𝑧 ∈ 𝑣) → 𝜑)
167	4	ad2antrr 727	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ 𝑧 ∈ 𝑣) → 𝐽 ∈ Top)
168		simplr 769	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ 𝑧 ∈ 𝑣) → 𝑣 ∈ 𝐽)
169	65	eltopss 22872	. . . . . . . . . . . . . . 15 ⊢ ((𝐽 ∈ Top ∧ 𝑣 ∈ 𝐽) → 𝑣 ⊆ 𝐶)
170	167, 168, 169	syl2anc 585	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ 𝑧 ∈ 𝑣) → 𝑣 ⊆ 𝐶)
171		simpr 484	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ 𝑧 ∈ 𝑣) → 𝑧 ∈ 𝑣)
172	170, 171	sseldd 3922	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ 𝑧 ∈ 𝑣) → 𝑧 ∈ 𝐶)
173		fvexd 6855	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ 𝑧 ∈ 𝑣) → ((nei‘𝐽)‘{𝑧}) ∈ V)
174	70	ad2antrr 727	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ 𝑧 ∈ 𝑣) → 𝐴 ∈ V)
175		opnneip 23084	. . . . . . . . . . . . . . . 16 ⊢ ((𝐽 ∈ Top ∧ 𝑣 ∈ 𝐽 ∧ 𝑧 ∈ 𝑣) → 𝑣 ∈ ((nei‘𝐽)‘{𝑧}))
176	4, 175	syl3an1 1164	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐽 ∧ 𝑧 ∈ 𝑣) → 𝑣 ∈ ((nei‘𝐽)‘{𝑧}))
177	176	3expa 1119	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ 𝑧 ∈ 𝑣) → 𝑣 ∈ ((nei‘𝐽)‘{𝑧}))
178		elrestr 17391	. . . . . . . . . . . . . 14 ⊢ ((((nei‘𝐽)‘{𝑧}) ∈ V ∧ 𝐴 ∈ V ∧ 𝑣 ∈ ((nei‘𝐽)‘{𝑧})) → (𝑣 ∩ 𝐴) ∈ (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴))
179	173, 174, 177, 178	syl3anc 1374	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ 𝑧 ∈ 𝑣) → (𝑣 ∩ 𝐴) ∈ (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴))
180	65, 42, 4, 32, 36, 69, 76, 86	cnextfvval 24030	. . . . . . . . . . . . . . 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 2822	. . . . . . . . . . . . . . . . . . . . 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 23857	. . . . . . . . . . . . . . . . . . . . 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 1374	. . . . . . . . . . . . . . . . . . . 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 2819	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐶 ↔ 𝑧 ∈ 𝐶))
193	192	anbi2d 631	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = 𝑧 → ((𝜑 ∧ 𝑥 ∈ 𝐶) ↔ (𝜑 ∧ 𝑧 ∈ 𝐶)))
194		sneq 4577	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑥 = 𝑧 → {𝑥} = {𝑧})
195	194	fveq2d 6844	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥 = 𝑧 → ((nei‘𝐽)‘{𝑥}) = ((nei‘𝐽)‘{𝑧}))
196	195	oveq1d 7382	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 = 𝑧 → (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴) = (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴))
197	196	oveq2d 7383	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = 𝑧 → (𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴)) = (𝐾 fLimf (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴)))
198	197	fveq1d 6842	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑧 → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾_t 𝐴))‘𝐹) = ((𝐾 fLimf (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴))‘𝐹))
199	198	neeq1d 2991	. . . . . . . . . . . . . . . . . . . . 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 1991	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴))‘𝐹) ≠ ∅)
202	42	hausflf2 23963	. . . . . . . . . . . . . . . . . . 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 1376	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴))‘𝐹) ≈ 1_o)
204		en1b 8972	. . . . . . . . . . . . . . . . . 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 23955	. . . . . . . . . . . . . . . . . . 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 1374	. . . . . . . . . . . . . . . . . 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 7696	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ∪ 𝐾 ∈ V)
212	211	ad2antrr 727	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐶) ∧ (𝑣 ∩ 𝐴) ∈ (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴)) → ∪ 𝐾 ∈ V)
213	42, 212	eqeltrid 2840	. . . . . . . . . . . . . . . . . . 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 23813	. . . . . . . . . . . . . . . . . . . 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 727	. . . . . . . . . . . . . . . . . . 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 23840	. . . . . . . . . . . . . . . . . . . . . 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 2839	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐶) ∧ (𝑣 ∩ 𝐴) ∈ (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴)) → (𝑣 ∩ 𝐴) ∈ (𝐴filGen(((nei‘𝐽)‘{𝑧}) ↾_t 𝐴)))
223		eqid 2736	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐴filGen(((nei‘𝐽)‘{𝑧}) ↾_t 𝐴)) = (𝐴filGen(((nei‘𝐽)‘{𝑧}) ↾_t 𝐴))
224	223	imaelfm 23916	. . . . . . . . . . . . . . . . . . 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 1376	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐶) ∧ (𝑣 ∩ 𝐴) ∈ (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴)) → (𝐹 “ (𝑣 ∩ 𝐴)) ∈ ((𝐵 FilMap 𝐹)‘(((nei‘𝐽)‘{𝑧}) ↾_t 𝐴)))
226		flimclsi 23943	. . . . . . . . . . . . . . . . . 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 3956	. . . . . . . . . . . . . . . 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 3957	. . . . . . . . . . . . . . 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 6853	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐾 fLimf (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴))‘𝐹) ∈ V
231	230	uniex 7695	. . . . . . . . . . . . . . . 16 ⊢ ∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑧}) ↾_t 𝐴))‘𝐹) ∈ V
232	231	snss 4728	. . . . . . . . . . . . . . 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 2836	. . . . . . . . . . . . 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 838	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ 𝑧 ∈ 𝑣) → (((𝐽CnExt𝐾)‘𝐹)‘𝑧) ∈ ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))))
236	235	adantlr 716	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))) ⊆ 𝑤) ∧ 𝑧 ∈ 𝑣) → (((𝐽CnExt𝐾)‘𝐹)‘𝑧) ∈ ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))))
237	165, 236	sseldd 3922	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))) ⊆ 𝑤) ∧ 𝑧 ∈ 𝑣) → (((𝐽CnExt𝐾)‘𝐹)‘𝑧) ∈ 𝑤)
238	237	ralrimiva 3129	. . . . . . . . 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 3152	. . . . . . 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 727	. . . . . 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 24031	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐽CnExt𝐾)‘𝐹):𝐶⟶𝐵)
244	243	ffund 6672	. . . . . . . . 9 ⊢ (𝜑 → Fun ((𝐽CnExt𝐾)‘𝐹))
245	244	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ ((nei‘𝐽)‘{𝑥})) → Fun ((𝐽CnExt𝐾)‘𝐹))
246	65	neii1 23071	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝑣 ∈ ((nei‘𝐽)‘{𝑥})) → 𝑣 ⊆ 𝐶)
247	4, 246	sylan 581	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ ((nei‘𝐽)‘{𝑥})) → 𝑣 ⊆ 𝐶)
248	243	fdmd 6678	. . . . . . . . . 10 ⊢ (𝜑 → dom ((𝐽CnExt𝐾)‘𝐹) = 𝐶)
249	248	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ ((nei‘𝐽)‘{𝑥})) → dom ((𝐽CnExt𝐾)‘𝐹) = 𝐶)
250	247, 249	sseqtrrd 3959	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ ((nei‘𝐽)‘{𝑥})) → 𝑣 ⊆ dom ((𝐽CnExt𝐾)‘𝐹))
251		funimass4 6904	. . . . . . . 8 ⊢ ((Fun ((𝐽CnExt𝐾)‘𝐹) ∧ 𝑣 ⊆ dom ((𝐽CnExt𝐾)‘𝐹)) → ((((𝐽CnExt𝐾)‘𝐹) “ 𝑣) ⊆ 𝑤 ↔ ∀𝑧 ∈ 𝑣 (((𝐽CnExt𝐾)‘𝐹)‘𝑧) ∈ 𝑤))
252	245, 250, 251	syl2anc 585	. . . . . . 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 3150	. . . . 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 3129	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ∀𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})∃𝑣 ∈ ((nei‘𝐽)‘{𝑥})(((𝐽CnExt𝐾)‘𝐹) “ 𝑣) ⊆ 𝑤)
257	256	ralrimiva 3129	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐶 ∀𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})∃𝑣 ∈ ((nei‘𝐽)‘{𝑥})(((𝐽CnExt𝐾)‘𝐹) “ 𝑣) ⊆ 𝑤)
258	65, 42	cnnei 23247	. . 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 1374	. 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 1087 = wceq 1542 ∈ wcel 2114 ≠ wne 2932 ∀wral 3051 ∃wrex 3061 Vcvv 3429 ∩ cin 3888 ⊆ wss 3889 ∅c0 4273 {csn 4567 ∪ cuni 4850 class class class wbr 5085 dom cdm 5631 ran crn 5632 “ cima 5634 Fun wfun 6492 ⟶wf 6494 ‘cfv 6498 (class class class)co 7367 1_oc1o 8398 ≈ cen 8890 ↾_t crest 17383 fBascfbas 21340 filGencfg 21341 Topctop 22858 TopOnctopon 22875 clsccl 22983 neicnei 23062 Cn ccn 23189 Hauscha 23273 Regcreg 23274 Filcfil 23810 FilMap cfm 23898 fLim cflim 23899 fLimf cflf 23900 CnExtccnext 24024

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-iin 4936 df-br 5086 df-opab 5148 df-mpt 5167 df-id 5526 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-ov 7370 df-oprab 7371 df-mpo 7372 df-1st 7942 df-2nd 7943 df-1o 8405 df-map 8775 df-pm 8776 df-en 8894 df-rest 17385 df-topgen 17406 df-fbas 21349 df-fg 21350 df-top 22859 df-topon 22876 df-cld 22984 df-ntr 22985 df-cls 22986 df-nei 23063 df-cn 23192 df-cnp 23193 df-haus 23280 df-reg 23281 df-fil 23811 df-fm 23903 df-flim 23904 df-flf 23905 df-cnext 24025

This theorem is referenced by: cnextucn 24267

W3C validator