Theorem flimfnfcls 24340

Description: A filter converges to a point iff every finer filter clusters there. Along with fclsfnflim 24339, 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 24338	. . . . 5 ⊢ (𝐽 fLim 𝑔) ⊆ (𝐽 fClus 𝑔)
2		flimtop 24277	. . . . . . . . 9 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐽 ∈ Top)
3		flimfnfcls.x	. . . . . . . . . 10 ⊢ 𝑋 = ∪ 𝐽
4	3	toptopon 23228	. . . . . . . . 9 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
5	2, 4	sylib 221	. . . . . . . 8 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐽 ∈ (TopOn‘𝑋))
6	5	ad2antrr 739	. . . . . . 7 ⊢ (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑔 ∈ (Fil‘𝑋)) ∧ 𝐹 ⊆ 𝑔) → 𝐽 ∈ (TopOn‘𝑋))
7		simplr 781	. . . . . . 7 ⊢ (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑔 ∈ (Fil‘𝑋)) ∧ 𝐹 ⊆ 𝑔) → 𝑔 ∈ (Fil‘𝑋))
8		simpr 490	. . . . . . 7 ⊢ (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑔 ∈ (Fil‘𝑋)) ∧ 𝐹 ⊆ 𝑔) → 𝐹 ⊆ 𝑔)
9		flimss2 24284	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑔 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝑔) → (𝐽 fLim 𝐹) ⊆ (𝐽 fLim 𝑔))
10	6, 7, 8, 9	syl3anc 1398	. . . . . 6 ⊢ (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑔 ∈ (Fil‘𝑋)) ∧ 𝐹 ⊆ 𝑔) → (𝐽 fLim 𝐹) ⊆ (𝐽 fLim 𝑔))
11		simpll 779	. . . . . 6 ⊢ (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑔 ∈ (Fil‘𝑋)) ∧ 𝐹 ⊆ 𝑔) → 𝐴 ∈ (𝐽 fLim 𝐹))
12	10, 11	sseldd 3932	. . . . 5 ⊢ (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑔 ∈ (Fil‘𝑋)) ∧ 𝐹 ⊆ 𝑔) → 𝐴 ∈ (𝐽 fLim 𝑔))
13	1, 12	sselid 3929	. . . 4 ⊢ (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑔 ∈ (Fil‘𝑋)) ∧ 𝐹 ⊆ 𝑔) → 𝐴 ∈ (𝐽 fClus 𝑔))
14	13	ex 418	. . 3 ⊢ ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑔 ∈ (Fil‘𝑋)) → (𝐹 ⊆ 𝑔 → 𝐴 ∈ (𝐽 fClus 𝑔)))
15	14	ralrimiva 3155	. 2 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → ∀𝑔 ∈ (Fil‘𝑋)(𝐹 ⊆ 𝑔 → 𝐴 ∈ (𝐽 fClus 𝑔)))
16		sseq2 3957	. . . . . 6 ⊢ (𝑔 = 𝐹 → (𝐹 ⊆ 𝑔 ↔ 𝐹 ⊆ 𝐹))
17		oveq2 7426	. . . . . . 7 ⊢ (𝑔 = 𝐹 → (𝐽 fClus 𝑔) = (𝐽 fClus 𝐹))
18	17	eleq2d 2847	. . . . . 6 ⊢ (𝑔 = 𝐹 → (𝐴 ∈ (𝐽 fClus 𝑔) ↔ 𝐴 ∈ (𝐽 fClus 𝐹)))
19	16, 18	imbi12d 347	. . . . 5 ⊢ (𝑔 = 𝐹 → ((𝐹 ⊆ 𝑔 → 𝐴 ∈ (𝐽 fClus 𝑔)) ↔ (𝐹 ⊆ 𝐹 → 𝐴 ∈ (𝐽 fClus 𝐹))))
20	19	rspcv 3573	. . . 4 ⊢ (𝐹 ∈ (Fil‘𝑋) → (∀𝑔 ∈ (Fil‘𝑋)(𝐹 ⊆ 𝑔 → 𝐴 ∈ (𝐽 fClus 𝑔)) → (𝐹 ⊆ 𝐹 → 𝐴 ∈ (𝐽 fClus 𝐹))))
21		ssid 3953	. . . . . 6 ⊢ 𝐹 ⊆ 𝐹
22		id 23	. . . . . 6 ⊢ ((𝐹 ⊆ 𝐹 → 𝐴 ∈ (𝐽 fClus 𝐹)) → (𝐹 ⊆ 𝐹 → 𝐴 ∈ (𝐽 fClus 𝐹)))
23	21, 22	mpi 21	. . . . 5 ⊢ ((𝐹 ⊆ 𝐹 → 𝐴 ∈ (𝐽 fClus 𝐹)) → 𝐴 ∈ (𝐽 fClus 𝐹))
24		fclstop 24323	. . . . . 6 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐽 ∈ Top)
25	3	fclselbas 24328	. . . . . 6 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐴 ∈ 𝑋)
26	24, 25	jca 521	. . . . 5 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋))
27	23, 26	syl 18	. . . 4 ⊢ ((𝐹 ⊆ 𝐹 → 𝐴 ∈ (𝐽 fClus 𝐹)) → (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋))
28	20, 27	syl6 36	. . 3 ⊢ (𝐹 ∈ (Fil‘𝑋) → (∀𝑔 ∈ (Fil‘𝑋)(𝐹 ⊆ 𝑔 → 𝐴 ∈ (𝐽 fClus 𝑔)) → (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)))
29		disjdif 4426	. . . . . . . . . . . . . 14 ⊢ (𝑜 ∩ (𝑋 ∖ 𝑜)) = ∅
30		simpll 779	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → 𝐹 ∈ (Fil‘𝑋))
31		simplrl 789	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → 𝐽 ∈ Top)
32	3	topopn 23217	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽)
33	31, 32	syl 18	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → 𝑋 ∈ 𝐽)
34		pwexg 5340	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑋 ∈ 𝐽 → 𝒫 𝑋 ∈ V)
35		rabexg 5299	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝒫 𝑋 ∈ V → {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥} ∈ V)
36	33, 34, 35	3syl 19	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥} ∈ V)
37		unexg 7758	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥} ∈ V) → (𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥}) ∈ V)
38	30, 36, 37	syl2anc 596	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → (𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥}) ∈ V)
39		ssfii 9404	. . . . . . . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . . . . . 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 24163	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ⊆ 𝒫 𝑋)
42		ssrab2 4028	. . . . . . . . . . . . . . . . . . . . . . . 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 4138	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥}) ⊆ 𝒫 𝑋)
45	44	ad2antrr 739	. . . . . . . . . . . . . . . . . . . . 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 4125	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥} ⊆ (𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥})
47		sseq2 3957	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 = (𝑋 ∖ 𝑜) → ((𝑋 ∖ 𝑜) ⊆ 𝑥 ↔ (𝑋 ∖ 𝑜) ⊆ (𝑋 ∖ 𝑜)))
48		difss 4083	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑋 ∖ 𝑜) ⊆ 𝑋
49		elpw2g 5295	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑋 ∈ 𝐽 → ((𝑋 ∖ 𝑜) ∈ 𝒫 𝑋 ↔ (𝑋 ∖ 𝑜) ⊆ 𝑋))
50	33, 49	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . 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 261	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → (𝑋 ∖ 𝑜) ∈ 𝒫 𝑋)
52		ssid 3953	. . . . . . . . . . . . . . . . . . . . . . . . 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 3647	. . . . . . . . . . . . . . . . . . . . . . 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 3929	. . . . . . . . . . . . . . . . . . . . . 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 4288	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → (𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥}) ≠ ∅)
57		sseq2 3957	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑥 = 𝑧 → ((𝑋 ∖ 𝑜) ⊆ 𝑥 ↔ (𝑋 ∖ 𝑜) ⊆ 𝑧))
58	57	elrab 3645	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑧 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥} ↔ (𝑧 ∈ 𝒫 𝑋 ∧ (𝑋 ∖ 𝑜) ⊆ 𝑧))
59	58	simprbi 503	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑧 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥} → (𝑋 ∖ 𝑜) ⊆ 𝑧)
60	59	ad2antll 742	. . . . . . . . . . . . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . . . . . . . . . . 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 794	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → ¬ 𝑜 ∈ 𝐹)
64	63	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . 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 4322	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑦 ∩ 𝑋) ⊆ 𝑜 ↔ (𝑦 ∩ (𝑋 ∖ 𝑜)) = ∅)
66		simplll 787	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 783	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 24164	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑦 ∈ 𝐹) → 𝑦 ⊆ 𝑋)
69	66, 67, 68	syl2anc 596	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3917	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑦 ⊆ 𝑋 ↔ (𝑦 ∩ 𝑋) = 𝑦)
71	69, 70	sylib 221	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3962	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 739	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 789	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4899	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑜 ∈ 𝐽 → 𝑜 ⊆ ∪ 𝐽)
76	75, 3	sseqtrrdi 3972	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑜 ∈ 𝐽 → 𝑜 ⊆ 𝑋)
77	76	ad2antrl 741	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → 𝑜 ⊆ 𝑋)
78	77	ad2antrr 739	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 490	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 24165	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑦 ∈ 𝐹 ∧ 𝑜 ⊆ 𝑋 ∧ 𝑦 ⊆ 𝑜)) → 𝑜 ∈ 𝐹)
81	73, 74, 78, 79, 80	syl13anc 1399	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 418	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 243	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 246	. . . . . . . . . . . . . . . . . . . . . . . . . 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 2970	. . . . . . . . . . . . . . . . . . . . . . . . 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 16	. . . . . . . . . . . . . . . . . . . . . . . 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 4355	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑦 ∩ (𝑋 ∖ 𝑜)) ⊆ (𝑦 ∩ 𝑧) ∧ (𝑦 ∩ (𝑋 ∖ 𝑜)) ≠ ∅) → (𝑦 ∩ 𝑧) ≠ ∅)
88	62, 86, 87	syl2anc 596	. . . . . . . . . . . . . . . . . . . . . . 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 3206	. . . . . . . . . . . . . . . . . . . . . 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 24160	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋))
91	30, 90	syl 18	. . . . . . . . . . . . . . . . . . . . . . 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 24167	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝑋 ∈ 𝐹)
94	30, 93	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → 𝑋 ∈ 𝐹)
95		eleq1 2849	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑜 = 𝑋 → (𝑜 ∈ 𝐹 ↔ 𝑋 ∈ 𝐹))
96	94, 95	syl5ibrcom 250	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → (𝑜 = 𝑋 → 𝑜 ∈ 𝐹))
97	96	necon3bd 2970	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → (¬ 𝑜 ∈ 𝐹 → 𝑜 ≠ 𝑋))
98	63, 97	mpd 16	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → 𝑜 ≠ 𝑋)
99		pssdifn0 4316	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑜 ⊆ 𝑋 ∧ 𝑜 ≠ 𝑋) → (𝑋 ∖ 𝑜) ≠ ∅)
100	77, 98, 99	syl2anc 596	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → (𝑋 ∖ 𝑜) ≠ ∅)
101		supfil 24207	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑋 ∈ 𝐽 ∧ (𝑋 ∖ 𝑜) ⊆ 𝑋 ∧ (𝑋 ∖ 𝑜) ≠ ∅) → {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥} ∈ (Fil‘𝑋))
102	33, 92, 100, 101	syl3anc 1398	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥} ∈ (Fil‘𝑋))
103		filfbas 24160	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ({𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥} ∈ (Fil‘𝑋) → {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥} ∈ (fBas‘𝑋))
104	102, 103	syl 18	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥} ∈ (fBas‘𝑋))
105		fbunfip 24181	. . . . . . . . . . . . . . . . . . . . . . 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 596	. . . . . . . . . . . . . . . . . . . . . 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 260	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → ¬ ∅ ∈ (fi‘(𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥})))
108		fsubbas 24179	. . . . . . . . . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . . . . . . . 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 1361	. . . . . . . . . . . . . . . . . . . 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 24184	. . . . . . . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . . . . . 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 3941	. . . . . . . . . . . . . . . . . 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 4139	. . . . . . . . . . . . . . . . 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 24190	. . . . . . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . . . . 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 3957	. . . . . . . . . . . . . . . . . . . 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 7426	. . . . . . . . . . . . . . . . . . . . 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 2847	. . . . . . . . . . . . . . . . . . . 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 347	. . . . . . . . . . . . . . . . . . 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 3573	. . . . . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . . . 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 45	. . . . . . . . . . . . . . . 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 490	. . . . . . . . . . . . . . . . . 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 789	. . . . . . . . . . . . . . . . . 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 793	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → 𝐴 ∈ 𝑜)
127	126	adantr 486	. . . . . . . . . . . . . . . . . 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 3932	. . . . . . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . . . 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 24327	. . . . . . . . . . . . . . . . . 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 1399	. . . . . . . . . . . . . . . . 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 418	. . . . . . . . . . . . . . . 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 48	. . . . . . . . . . . . . . 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 2972	. . . . . . . . . . . . . 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 21	. . . . . . . . . . . . 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 473	. . . . . . . . . . . 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 462	. . . . . . . . . . 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 116	. . . . . . . . . 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 418	. . . . . . . . 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 87	. . . . . . . 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 3163	. . . . . . 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 785	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) → 𝐴 ∈ 𝑋)
143	141, 142	jctild 535	. . . . . 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 783	. . . . . . . 8 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) → 𝐽 ∈ Top)
145	144, 4	sylib 221	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) → 𝐽 ∈ (TopOn‘𝑋))
146		simpl 488	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) → 𝐹 ∈ (Fil‘𝑋))
147		flimopn 24287	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fLim 𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → 𝑜 ∈ 𝐹))))
148	145, 146, 147	syl2anc 596	. . . . . 6 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) → (𝐴 ∈ (𝐽 fLim 𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → 𝑜 ∈ 𝐹))))
149	143, 148	sylibrd 262	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) → (∀𝑔 ∈ (Fil‘𝑋)(𝐹 ⊆ 𝑔 → 𝐴 ∈ (𝐽 fClus 𝑔)) → 𝐴 ∈ (𝐽 fLim 𝐹)))
150	149	ex 418	. . . 4 ⊢ (𝐹 ∈ (Fil‘𝑋) → ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋) → (∀𝑔 ∈ (Fil‘𝑋)(𝐹 ⊆ 𝑔 → 𝐴 ∈ (𝐽 fClus 𝑔)) → 𝐴 ∈ (𝐽 fLim 𝐹))))
151	150	com23 87	. . 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 44	. 2 ⊢ (𝐹 ∈ (Fil‘𝑋) → (∀𝑔 ∈ (Fil‘𝑋)(𝐹 ⊆ 𝑔 → 𝐴 ∈ (𝐽 fClus 𝑔)) → 𝐴 ∈ (𝐽 fLim 𝐹)))
153	15, 152	impbid2 229	1 ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝐴 ∈ (𝐽 fLim 𝐹) ↔ ∀𝑔 ∈ (Fil‘𝑋)(𝐹 ⊆ 𝑔 → 𝐴 ∈ (𝐽 fClus 𝑔))))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2956  ∀wral 3077  {crab 3413  Vcvv 3451   ∖ cdif 3896   ∪ cun 3897   ∩ cin 3898   ⊆ wss 3899  ∅c0 4279  𝒫 cpw 4557  ∪ cuni 4867  ‘cfv 6537  (class class class)co 7418  ficfi 9395  fBascfbas 21659  filGencfg 21660  Topctop 23204  TopOnctopon 23221  Filcfil 24157   fLim cflim 24246   fClus cfcls 24248

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-iin 4954  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7421  df-oprab 7422  df-mpo 7423  df-om 7876  df-1o 8469  df-2o 8470  df-en 8967  df-fin 8970  df-fi 9396  df-fbas 21668  df-fg 21669  df-top 23205  df-topon 23222  df-cld 23330  df-ntr 23331  df-cls 23332  df-nei 23409  df-fil 24158  df-flim 24251  df-fcls 24253

This theorem is used by:  cnpfcf  24353