Theorem flimfnfcls 23938

Description: A filter converges to a point iff every finer filter clusters there. Along with fclsfnflim 23937, this theorem illustrates the duality between convergence and clustering. (Contributed by Jeff Hankins, 12-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)

Hypothesis

Ref	Expression
flimfnfcls.x	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
flimfnfcls	⊢ (𝐹 ∈ (Fil‘𝑋) → (𝐴 ∈ (𝐽 fLim 𝐹) ↔ ∀𝑔 ∈ (Fil‘𝑋)(𝐹 ⊆ 𝑔 → 𝐴 ∈ (𝐽 fClus 𝑔))))

Distinct variable groups:   𝐴,𝑔   𝑔,𝐹   𝑔,𝐽   𝑔,𝑋

Proof of Theorem flimfnfcls

Dummy variables 𝑜 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		flimfcls 23936	. . . . 5 ⊢ (𝐽 fLim 𝑔) ⊆ (𝐽 fClus 𝑔)
2		flimtop 23875	. . . . . . . . 9 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐽 ∈ Top)
3		flimfnfcls.x	. . . . . . . . . 10 ⊢ 𝑋 = ∪ 𝐽
4	3	toptopon 22827	. . . . . . . . 9 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
5	2, 4	sylib 218	. . . . . . . 8 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐽 ∈ (TopOn‘𝑋))
6	5	ad2antrr 726	. . . . . . 7 ⊢ (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑔 ∈ (Fil‘𝑋)) ∧ 𝐹 ⊆ 𝑔) → 𝐽 ∈ (TopOn‘𝑋))
7		simplr 768	. . . . . . 7 ⊢ (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑔 ∈ (Fil‘𝑋)) ∧ 𝐹 ⊆ 𝑔) → 𝑔 ∈ (Fil‘𝑋))
8		simpr 484	. . . . . . 7 ⊢ (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑔 ∈ (Fil‘𝑋)) ∧ 𝐹 ⊆ 𝑔) → 𝐹 ⊆ 𝑔)
9		flimss2 23882	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑔 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝑔) → (𝐽 fLim 𝐹) ⊆ (𝐽 fLim 𝑔))
10	6, 7, 8, 9	syl3anc 1373	. . . . . 6 ⊢ (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑔 ∈ (Fil‘𝑋)) ∧ 𝐹 ⊆ 𝑔) → (𝐽 fLim 𝐹) ⊆ (𝐽 fLim 𝑔))
11		simpll 766	. . . . . 6 ⊢ (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑔 ∈ (Fil‘𝑋)) ∧ 𝐹 ⊆ 𝑔) → 𝐴 ∈ (𝐽 fLim 𝐹))
12	10, 11	sseldd 3930	. . . . 5 ⊢ (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑔 ∈ (Fil‘𝑋)) ∧ 𝐹 ⊆ 𝑔) → 𝐴 ∈ (𝐽 fLim 𝑔))
13	1, 12	sselid 3927	. . . 4 ⊢ (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑔 ∈ (Fil‘𝑋)) ∧ 𝐹 ⊆ 𝑔) → 𝐴 ∈ (𝐽 fClus 𝑔))
14	13	ex 412	. . 3 ⊢ ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑔 ∈ (Fil‘𝑋)) → (𝐹 ⊆ 𝑔 → 𝐴 ∈ (𝐽 fClus 𝑔)))
15	14	ralrimiva 3124	. 2 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → ∀𝑔 ∈ (Fil‘𝑋)(𝐹 ⊆ 𝑔 → 𝐴 ∈ (𝐽 fClus 𝑔)))
16		sseq2 3956	. . . . . 6 ⊢ (𝑔 = 𝐹 → (𝐹 ⊆ 𝑔 ↔ 𝐹 ⊆ 𝐹))
17		oveq2 7349	. . . . . . 7 ⊢ (𝑔 = 𝐹 → (𝐽 fClus 𝑔) = (𝐽 fClus 𝐹))
18	17	eleq2d 2817	. . . . . 6 ⊢ (𝑔 = 𝐹 → (𝐴 ∈ (𝐽 fClus 𝑔) ↔ 𝐴 ∈ (𝐽 fClus 𝐹)))
19	16, 18	imbi12d 344	. . . . 5 ⊢ (𝑔 = 𝐹 → ((𝐹 ⊆ 𝑔 → 𝐴 ∈ (𝐽 fClus 𝑔)) ↔ (𝐹 ⊆ 𝐹 → 𝐴 ∈ (𝐽 fClus 𝐹))))
20	19	rspcv 3568	. . . 4 ⊢ (𝐹 ∈ (Fil‘𝑋) → (∀𝑔 ∈ (Fil‘𝑋)(𝐹 ⊆ 𝑔 → 𝐴 ∈ (𝐽 fClus 𝑔)) → (𝐹 ⊆ 𝐹 → 𝐴 ∈ (𝐽 fClus 𝐹))))
21		ssid 3952	. . . . . 6 ⊢ 𝐹 ⊆ 𝐹
22		id 22	. . . . . 6 ⊢ ((𝐹 ⊆ 𝐹 → 𝐴 ∈ (𝐽 fClus 𝐹)) → (𝐹 ⊆ 𝐹 → 𝐴 ∈ (𝐽 fClus 𝐹)))
23	21, 22	mpi 20	. . . . 5 ⊢ ((𝐹 ⊆ 𝐹 → 𝐴 ∈ (𝐽 fClus 𝐹)) → 𝐴 ∈ (𝐽 fClus 𝐹))
24		fclstop 23921	. . . . . 6 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐽 ∈ Top)
25	3	fclselbas 23926	. . . . . 6 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐴 ∈ 𝑋)
26	24, 25	jca 511	. . . . 5 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋))
27	23, 26	syl 17	. . . 4 ⊢ ((𝐹 ⊆ 𝐹 → 𝐴 ∈ (𝐽 fClus 𝐹)) → (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋))
28	20, 27	syl6 35	. . 3 ⊢ (𝐹 ∈ (Fil‘𝑋) → (∀𝑔 ∈ (Fil‘𝑋)(𝐹 ⊆ 𝑔 → 𝐴 ∈ (𝐽 fClus 𝑔)) → (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)))
29		disjdif 4417	. . . . . . . . . . . . . 14 ⊢ (𝑜 ∩ (𝑋 ∖ 𝑜)) = ∅
30		simpll 766	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → 𝐹 ∈ (Fil‘𝑋))
31		simplrl 776	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → 𝐽 ∈ Top)
32	3	topopn 22816	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽)
33	31, 32	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → 𝑋 ∈ 𝐽)
34		pwexg 5311	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑋 ∈ 𝐽 → 𝒫 𝑋 ∈ V)
35		rabexg 5270	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝒫 𝑋 ∈ V → {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥} ∈ V)
36	33, 34, 35	3syl 18	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥} ∈ V)
37		unexg 7671	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥} ∈ V) → (𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥}) ∈ V)
38	30, 36, 37	syl2anc 584	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → (𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥}) ∈ V)
39		ssfii 9298	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥}) ∈ V → (𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥}) ⊆ (fi‘(𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥})))
40	38, 39	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → (𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥}) ⊆ (fi‘(𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥})))
41		filsspw 23761	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ⊆ 𝒫 𝑋)
42		ssrab2 4025	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥} ⊆ 𝒫 𝑋
43	42	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐹 ∈ (Fil‘𝑋) → {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥} ⊆ 𝒫 𝑋)
44	41, 43	unssd 4137	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥}) ⊆ 𝒫 𝑋)
45	44	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → (𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥}) ⊆ 𝒫 𝑋)
46		ssun2 4124	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥} ⊆ (𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥})
47		sseq2 3956	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 = (𝑋 ∖ 𝑜) → ((𝑋 ∖ 𝑜) ⊆ 𝑥 ↔ (𝑋 ∖ 𝑜) ⊆ (𝑋 ∖ 𝑜)))
48		difss 4081	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑋 ∖ 𝑜) ⊆ 𝑋
49		elpw2g 5266	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑋 ∈ 𝐽 → ((𝑋 ∖ 𝑜) ∈ 𝒫 𝑋 ↔ (𝑋 ∖ 𝑜) ⊆ 𝑋))
50	33, 49	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → ((𝑋 ∖ 𝑜) ∈ 𝒫 𝑋 ↔ (𝑋 ∖ 𝑜) ⊆ 𝑋))
51	48, 50	mpbiri 258	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → (𝑋 ∖ 𝑜) ∈ 𝒫 𝑋)
52		ssid 3952	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑋 ∖ 𝑜) ⊆ (𝑋 ∖ 𝑜)
53	52	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → (𝑋 ∖ 𝑜) ⊆ (𝑋 ∖ 𝑜))
54	47, 51, 53	elrabd 3644	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → (𝑋 ∖ 𝑜) ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥})
55	46, 54	sselid 3927	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → (𝑋 ∖ 𝑜) ∈ (𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥}))
56	55	ne0d 4287	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → (𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥}) ≠ ∅)
57		sseq2 3956	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑥 = 𝑧 → ((𝑋 ∖ 𝑜) ⊆ 𝑥 ↔ (𝑋 ∖ 𝑜) ⊆ 𝑧))
58	57	elrab 3642	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑧 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥} ↔ (𝑧 ∈ 𝒫 𝑋 ∧ (𝑋 ∖ 𝑜) ⊆ 𝑧))
59	58	simprbi 496	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑧 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥} → (𝑋 ∖ 𝑜) ⊆ 𝑧)
60	59	ad2antll 729	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) ∧ (𝑦 ∈ 𝐹 ∧ 𝑧 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥})) → (𝑋 ∖ 𝑜) ⊆ 𝑧)
61		sslin 4188	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑋 ∖ 𝑜) ⊆ 𝑧 → (𝑦 ∩ (𝑋 ∖ 𝑜)) ⊆ (𝑦 ∩ 𝑧))
62	60, 61	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) ∧ (𝑦 ∈ 𝐹 ∧ 𝑧 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥})) → (𝑦 ∩ (𝑋 ∖ 𝑜)) ⊆ (𝑦 ∩ 𝑧))
63		simprrr 781	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → ¬ 𝑜 ∈ 𝐹)
64	63	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) ∧ (𝑦 ∈ 𝐹 ∧ 𝑧 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥})) → ¬ 𝑜 ∈ 𝐹)
65		inssdif0 4319	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑦 ∩ 𝑋) ⊆ 𝑜 ↔ (𝑦 ∩ (𝑋 ∖ 𝑜)) = ∅)
66		simplll 774	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) ∧ (𝑦 ∈ 𝐹 ∧ 𝑧 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥})) → 𝐹 ∈ (Fil‘𝑋))
67		simprl 770	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) ∧ (𝑦 ∈ 𝐹 ∧ 𝑧 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥})) → 𝑦 ∈ 𝐹)
68		filelss 23762	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑦 ∈ 𝐹) → 𝑦 ⊆ 𝑋)
69	66, 67, 68	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) ∧ (𝑦 ∈ 𝐹 ∧ 𝑧 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥})) → 𝑦 ⊆ 𝑋)
70		dfss2 3915	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑦 ⊆ 𝑋 ↔ (𝑦 ∩ 𝑋) = 𝑦)
71	69, 70	sylib 218	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) ∧ (𝑦 ∈ 𝐹 ∧ 𝑧 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥})) → (𝑦 ∩ 𝑋) = 𝑦)
72	71	sseq1d 3961	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) ∧ (𝑦 ∈ 𝐹 ∧ 𝑧 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥})) → ((𝑦 ∩ 𝑋) ⊆ 𝑜 ↔ 𝑦 ⊆ 𝑜))
73	30	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) ∧ (𝑦 ∈ 𝐹 ∧ 𝑧 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥})) ∧ 𝑦 ⊆ 𝑜) → 𝐹 ∈ (Fil‘𝑋))
74		simplrl 776	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) ∧ (𝑦 ∈ 𝐹 ∧ 𝑧 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥})) ∧ 𝑦 ⊆ 𝑜) → 𝑦 ∈ 𝐹)
75		elssuni 4884	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑜 ∈ 𝐽 → 𝑜 ⊆ ∪ 𝐽)
76	75, 3	sseqtrrdi 3971	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑜 ∈ 𝐽 → 𝑜 ⊆ 𝑋)
77	76	ad2antrl 728	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → 𝑜 ⊆ 𝑋)
78	77	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) ∧ (𝑦 ∈ 𝐹 ∧ 𝑧 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥})) ∧ 𝑦 ⊆ 𝑜) → 𝑜 ⊆ 𝑋)
79		simpr 484	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) ∧ (𝑦 ∈ 𝐹 ∧ 𝑧 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥})) ∧ 𝑦 ⊆ 𝑜) → 𝑦 ⊆ 𝑜)
80		filss 23763	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑦 ∈ 𝐹 ∧ 𝑜 ⊆ 𝑋 ∧ 𝑦 ⊆ 𝑜)) → 𝑜 ∈ 𝐹)
81	73, 74, 78, 79, 80	syl13anc 1374	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) ∧ (𝑦 ∈ 𝐹 ∧ 𝑧 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥})) ∧ 𝑦 ⊆ 𝑜) → 𝑜 ∈ 𝐹)
82	81	ex 412	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) ∧ (𝑦 ∈ 𝐹 ∧ 𝑧 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥})) → (𝑦 ⊆ 𝑜 → 𝑜 ∈ 𝐹))
83	72, 82	sylbid 240	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) ∧ (𝑦 ∈ 𝐹 ∧ 𝑧 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥})) → ((𝑦 ∩ 𝑋) ⊆ 𝑜 → 𝑜 ∈ 𝐹))
84	65, 83	biimtrrid 243	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) ∧ (𝑦 ∈ 𝐹 ∧ 𝑧 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥})) → ((𝑦 ∩ (𝑋 ∖ 𝑜)) = ∅ → 𝑜 ∈ 𝐹))
85	84	necon3bd 2942	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) ∧ (𝑦 ∈ 𝐹 ∧ 𝑧 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥})) → (¬ 𝑜 ∈ 𝐹 → (𝑦 ∩ (𝑋 ∖ 𝑜)) ≠ ∅))
86	64, 85	mpd 15	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) ∧ (𝑦 ∈ 𝐹 ∧ 𝑧 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥})) → (𝑦 ∩ (𝑋 ∖ 𝑜)) ≠ ∅)
87		ssn0 4349	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑦 ∩ (𝑋 ∖ 𝑜)) ⊆ (𝑦 ∩ 𝑧) ∧ (𝑦 ∩ (𝑋 ∖ 𝑜)) ≠ ∅) → (𝑦 ∩ 𝑧) ≠ ∅)
88	62, 86, 87	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) ∧ (𝑦 ∈ 𝐹 ∧ 𝑧 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥})) → (𝑦 ∩ 𝑧) ≠ ∅)
89	88	ralrimivva 3175	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → ∀𝑦 ∈ 𝐹 ∀𝑧 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥} (𝑦 ∩ 𝑧) ≠ ∅)
90		filfbas 23758	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋))
91	30, 90	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → 𝐹 ∈ (fBas‘𝑋))
92	48	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → (𝑋 ∖ 𝑜) ⊆ 𝑋)
93		filtop 23765	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝑋 ∈ 𝐹)
94	30, 93	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → 𝑋 ∈ 𝐹)
95		eleq1 2819	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑜 = 𝑋 → (𝑜 ∈ 𝐹 ↔ 𝑋 ∈ 𝐹))
96	94, 95	syl5ibrcom 247	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → (𝑜 = 𝑋 → 𝑜 ∈ 𝐹))
97	96	necon3bd 2942	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → (¬ 𝑜 ∈ 𝐹 → 𝑜 ≠ 𝑋))
98	63, 97	mpd 15	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → 𝑜 ≠ 𝑋)
99		pssdifn0 4313	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑜 ⊆ 𝑋 ∧ 𝑜 ≠ 𝑋) → (𝑋 ∖ 𝑜) ≠ ∅)
100	77, 98, 99	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → (𝑋 ∖ 𝑜) ≠ ∅)
101		supfil 23805	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑋 ∈ 𝐽 ∧ (𝑋 ∖ 𝑜) ⊆ 𝑋 ∧ (𝑋 ∖ 𝑜) ≠ ∅) → {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥} ∈ (Fil‘𝑋))
102	33, 92, 100, 101	syl3anc 1373	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥} ∈ (Fil‘𝑋))
103		filfbas 23758	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ({𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥} ∈ (Fil‘𝑋) → {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥} ∈ (fBas‘𝑋))
104	102, 103	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥} ∈ (fBas‘𝑋))
105		fbunfip 23779	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥} ∈ (fBas‘𝑋)) → (¬ ∅ ∈ (fi‘(𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥})) ↔ ∀𝑦 ∈ 𝐹 ∀𝑧 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥} (𝑦 ∩ 𝑧) ≠ ∅))
106	91, 104, 105	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → (¬ ∅ ∈ (fi‘(𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥})) ↔ ∀𝑦 ∈ 𝐹 ∀𝑧 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥} (𝑦 ∩ 𝑧) ≠ ∅))
107	89, 106	mpbird 257	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → ¬ ∅ ∈ (fi‘(𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥})))
108		fsubbas 23777	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑋 ∈ 𝐹 → ((fi‘(𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥})) ∈ (fBas‘𝑋) ↔ ((𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥}) ⊆ 𝒫 𝑋 ∧ (𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥}) ≠ ∅ ∧ ¬ ∅ ∈ (fi‘(𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥})))))
109	94, 108	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → ((fi‘(𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥})) ∈ (fBas‘𝑋) ↔ ((𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥}) ⊆ 𝒫 𝑋 ∧ (𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥}) ≠ ∅ ∧ ¬ ∅ ∈ (fi‘(𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥})))))
110	45, 56, 107, 109	mpbir3and 1343	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → (fi‘(𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥})) ∈ (fBas‘𝑋))
111		ssfg 23782	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((fi‘(𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥})) ∈ (fBas‘𝑋) → (fi‘(𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥})) ⊆ (𝑋filGen(fi‘(𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥}))))
112	110, 111	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → (fi‘(𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥})) ⊆ (𝑋filGen(fi‘(𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥}))))
113	40, 112	sstrd 3940	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → (𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥}) ⊆ (𝑋filGen(fi‘(𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥}))))
114	113	unssad 4138	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → 𝐹 ⊆ (𝑋filGen(fi‘(𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥}))))
115		fgcl 23788	. . . . . . . . . . . . . . . . . . 19 ⊢ ((fi‘(𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥})) ∈ (fBas‘𝑋) → (𝑋filGen(fi‘(𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥}))) ∈ (Fil‘𝑋))
116	110, 115	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → (𝑋filGen(fi‘(𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥}))) ∈ (Fil‘𝑋))
117		sseq2 3956	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑔 = (𝑋filGen(fi‘(𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥}))) → (𝐹 ⊆ 𝑔 ↔ 𝐹 ⊆ (𝑋filGen(fi‘(𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥})))))
118		oveq2 7349	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑔 = (𝑋filGen(fi‘(𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥}))) → (𝐽 fClus 𝑔) = (𝐽 fClus (𝑋filGen(fi‘(𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥})))))
119	118	eleq2d 2817	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑔 = (𝑋filGen(fi‘(𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥}))) → (𝐴 ∈ (𝐽 fClus 𝑔) ↔ 𝐴 ∈ (𝐽 fClus (𝑋filGen(fi‘(𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥}))))))
120	117, 119	imbi12d 344	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑔 = (𝑋filGen(fi‘(𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥}))) → ((𝐹 ⊆ 𝑔 → 𝐴 ∈ (𝐽 fClus 𝑔)) ↔ (𝐹 ⊆ (𝑋filGen(fi‘(𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥}))) → 𝐴 ∈ (𝐽 fClus (𝑋filGen(fi‘(𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥})))))))
121	120	rspcv 3568	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑋filGen(fi‘(𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥}))) ∈ (Fil‘𝑋) → (∀𝑔 ∈ (Fil‘𝑋)(𝐹 ⊆ 𝑔 → 𝐴 ∈ (𝐽 fClus 𝑔)) → (𝐹 ⊆ (𝑋filGen(fi‘(𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥}))) → 𝐴 ∈ (𝐽 fClus (𝑋filGen(fi‘(𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥})))))))
122	116, 121	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → (∀𝑔 ∈ (Fil‘𝑋)(𝐹 ⊆ 𝑔 → 𝐴 ∈ (𝐽 fClus 𝑔)) → (𝐹 ⊆ (𝑋filGen(fi‘(𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥}))) → 𝐴 ∈ (𝐽 fClus (𝑋filGen(fi‘(𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥})))))))
123	114, 122	mpid 44	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → (∀𝑔 ∈ (Fil‘𝑋)(𝐹 ⊆ 𝑔 → 𝐴 ∈ (𝐽 fClus 𝑔)) → 𝐴 ∈ (𝐽 fClus (𝑋filGen(fi‘(𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥}))))))
124		simpr 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) ∧ 𝐴 ∈ (𝐽 fClus (𝑋filGen(fi‘(𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥}))))) → 𝐴 ∈ (𝐽 fClus (𝑋filGen(fi‘(𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥})))))
125		simplrl 776	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) ∧ 𝐴 ∈ (𝐽 fClus (𝑋filGen(fi‘(𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥}))))) → 𝑜 ∈ 𝐽)
126		simprrl 780	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → 𝐴 ∈ 𝑜)
127	126	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) ∧ 𝐴 ∈ (𝐽 fClus (𝑋filGen(fi‘(𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥}))))) → 𝐴 ∈ 𝑜)
128	113, 55	sseldd 3930	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → (𝑋 ∖ 𝑜) ∈ (𝑋filGen(fi‘(𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥}))))
129	128	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) ∧ 𝐴 ∈ (𝐽 fClus (𝑋filGen(fi‘(𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥}))))) → (𝑋 ∖ 𝑜) ∈ (𝑋filGen(fi‘(𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥}))))
130		fclsopni 23925	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ (𝐽 fClus (𝑋filGen(fi‘(𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥})))) ∧ (𝑜 ∈ 𝐽 ∧ 𝐴 ∈ 𝑜 ∧ (𝑋 ∖ 𝑜) ∈ (𝑋filGen(fi‘(𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥}))))) → (𝑜 ∩ (𝑋 ∖ 𝑜)) ≠ ∅)
131	124, 125, 127, 129, 130	syl13anc 1374	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) ∧ 𝐴 ∈ (𝐽 fClus (𝑋filGen(fi‘(𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥}))))) → (𝑜 ∩ (𝑋 ∖ 𝑜)) ≠ ∅)
132	131	ex 412	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → (𝐴 ∈ (𝐽 fClus (𝑋filGen(fi‘(𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥})))) → (𝑜 ∩ (𝑋 ∖ 𝑜)) ≠ ∅))
133	123, 132	syld 47	. . . . . . . . . . . . . . 15 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → (∀𝑔 ∈ (Fil‘𝑋)(𝐹 ⊆ 𝑔 → 𝐴 ∈ (𝐽 fClus 𝑔)) → (𝑜 ∩ (𝑋 ∖ 𝑜)) ≠ ∅))
134	133	necon2bd 2944	. . . . . . . . . . . . . 14 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → ((𝑜 ∩ (𝑋 ∖ 𝑜)) = ∅ → ¬ ∀𝑔 ∈ (Fil‘𝑋)(𝐹 ⊆ 𝑔 → 𝐴 ∈ (𝐽 fClus 𝑔))))
135	29, 134	mpi 20	. . . . . . . . . . . . 13 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → ¬ ∀𝑔 ∈ (Fil‘𝑋)(𝐹 ⊆ 𝑔 → 𝐴 ∈ (𝐽 fClus 𝑔)))
136	135	anassrs 467	. . . . . . . . . . . 12 ⊢ ((((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ 𝑜 ∈ 𝐽) ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹)) → ¬ ∀𝑔 ∈ (Fil‘𝑋)(𝐹 ⊆ 𝑔 → 𝐴 ∈ (𝐽 fClus 𝑔)))
137	136	expr 456	. . . . . . . . . . 11 ⊢ ((((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ 𝑜 ∈ 𝐽) ∧ 𝐴 ∈ 𝑜) → (¬ 𝑜 ∈ 𝐹 → ¬ ∀𝑔 ∈ (Fil‘𝑋)(𝐹 ⊆ 𝑔 → 𝐴 ∈ (𝐽 fClus 𝑔))))
138	137	con4d 115	. . . . . . . . . 10 ⊢ ((((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ 𝑜 ∈ 𝐽) ∧ 𝐴 ∈ 𝑜) → (∀𝑔 ∈ (Fil‘𝑋)(𝐹 ⊆ 𝑔 → 𝐴 ∈ (𝐽 fClus 𝑔)) → 𝑜 ∈ 𝐹))
139	138	ex 412	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ 𝑜 ∈ 𝐽) → (𝐴 ∈ 𝑜 → (∀𝑔 ∈ (Fil‘𝑋)(𝐹 ⊆ 𝑔 → 𝐴 ∈ (𝐽 fClus 𝑔)) → 𝑜 ∈ 𝐹)))
140	139	com23 86	. . . . . . . 8 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ 𝑜 ∈ 𝐽) → (∀𝑔 ∈ (Fil‘𝑋)(𝐹 ⊆ 𝑔 → 𝐴 ∈ (𝐽 fClus 𝑔)) → (𝐴 ∈ 𝑜 → 𝑜 ∈ 𝐹)))
141	140	ralrimdva 3132	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) → (∀𝑔 ∈ (Fil‘𝑋)(𝐹 ⊆ 𝑔 → 𝐴 ∈ (𝐽 fClus 𝑔)) → ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → 𝑜 ∈ 𝐹)))
142		simprr 772	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) → 𝐴 ∈ 𝑋)
143	141, 142	jctild 525	. . . . . 6 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) → (∀𝑔 ∈ (Fil‘𝑋)(𝐹 ⊆ 𝑔 → 𝐴 ∈ (𝐽 fClus 𝑔)) → (𝐴 ∈ 𝑋 ∧ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → 𝑜 ∈ 𝐹))))
144		simprl 770	. . . . . . . 8 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) → 𝐽 ∈ Top)
145	144, 4	sylib 218	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) → 𝐽 ∈ (TopOn‘𝑋))
146		simpl 482	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) → 𝐹 ∈ (Fil‘𝑋))
147		flimopn 23885	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fLim 𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → 𝑜 ∈ 𝐹))))
148	145, 146, 147	syl2anc 584	. . . . . 6 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) → (𝐴 ∈ (𝐽 fLim 𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → 𝑜 ∈ 𝐹))))
149	143, 148	sylibrd 259	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) → (∀𝑔 ∈ (Fil‘𝑋)(𝐹 ⊆ 𝑔 → 𝐴 ∈ (𝐽 fClus 𝑔)) → 𝐴 ∈ (𝐽 fLim 𝐹)))
150	149	ex 412	. . . 4 ⊢ (𝐹 ∈ (Fil‘𝑋) → ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋) → (∀𝑔 ∈ (Fil‘𝑋)(𝐹 ⊆ 𝑔 → 𝐴 ∈ (𝐽 fClus 𝑔)) → 𝐴 ∈ (𝐽 fLim 𝐹))))
151	150	com23 86	. . 3 ⊢ (𝐹 ∈ (Fil‘𝑋) → (∀𝑔 ∈ (Fil‘𝑋)(𝐹 ⊆ 𝑔 → 𝐴 ∈ (𝐽 fClus 𝑔)) → ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ (𝐽 fLim 𝐹))))
152	28, 151	mpdd 43	. 2 ⊢ (𝐹 ∈ (Fil‘𝑋) → (∀𝑔 ∈ (Fil‘𝑋)(𝐹 ⊆ 𝑔 → 𝐴 ∈ (𝐽 fClus 𝑔)) → 𝐴 ∈ (𝐽 fLim 𝐹)))
153	15, 152	impbid2 226	1 ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝐴 ∈ (𝐽 fLim 𝐹) ↔ ∀𝑔 ∈ (Fil‘𝑋)(𝐹 ⊆ 𝑔 → 𝐴 ∈ (𝐽 fClus 𝑔))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111   ≠ wne 2928  ∀wral 3047  {crab 3395  Vcvv 3436   ∖ cdif 3894   ∪ cun 3895   ∩ cin 3896   ⊆ wss 3897  ∅c0 4278  𝒫 cpw 4545  ∪ cuni 4854  ‘cfv 6476  (class class class)co 7341  ficfi 9289  fBascfbas 21274  filGencfg 21275  Topctop 22803  TopOnctopon 22820  Filcfil 23755   fLim cflim 23844   fClus cfcls 23846

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 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5212  ax-sep 5229  ax-nul 5239  ax-pow 5298  ax-pr 5365  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-int 4893  df-iun 4938  df-iin 4939  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5506  df-eprel 5511  df-po 5519  df-so 5520  df-fr 5564  df-we 5566  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-ord 6304  df-on 6305  df-lim 6306  df-suc 6307  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-ov 7344  df-oprab 7345  df-mpo 7346  df-om 7792  df-1o 8380  df-2o 8381  df-en 8865  df-fin 8868  df-fi 9290  df-fbas 21283  df-fg 21284  df-top 22804  df-topon 22821  df-cld 22929  df-ntr 22930  df-cls 22931  df-nei 23008  df-fil 23756  df-flim 23849  df-fcls 23851

This theorem is referenced by:  cnpfcf  23951