Theorem flimfnfcls 23984

Description: A filter converges to a point iff every finer filter clusters there. Along with fclsfnflim 23983, 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 23982	. . . . 5 ⊢ (𝐽 fLim 𝑔) ⊆ (𝐽 fClus 𝑔)
2		flimtop 23921	. . . . . . . . 9 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐽 ∈ Top)
3		flimfnfcls.x	. . . . . . . . . 10 ⊢ 𝑋 = ∪ 𝐽
4	3	toptopon 22873	. . . . . . . . 9 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
5	2, 4	sylib 218	. . . . . . . 8 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐽 ∈ (TopOn‘𝑋))
6	5	ad2antrr 727	. . . . . . 7 ⊢ (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑔 ∈ (Fil‘𝑋)) ∧ 𝐹 ⊆ 𝑔) → 𝐽 ∈ (TopOn‘𝑋))
7		simplr 769	. . . . . . 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 23928	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑔 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝑔) → (𝐽 fLim 𝐹) ⊆ (𝐽 fLim 𝑔))
10	6, 7, 8, 9	syl3anc 1374	. . . . . 6 ⊢ (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑔 ∈ (Fil‘𝑋)) ∧ 𝐹 ⊆ 𝑔) → (𝐽 fLim 𝐹) ⊆ (𝐽 fLim 𝑔))
11		simpll 767	. . . . . 6 ⊢ (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑔 ∈ (Fil‘𝑋)) ∧ 𝐹 ⊆ 𝑔) → 𝐴 ∈ (𝐽 fLim 𝐹))
12	10, 11	sseldd 3936	. . . . 5 ⊢ (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑔 ∈ (Fil‘𝑋)) ∧ 𝐹 ⊆ 𝑔) → 𝐴 ∈ (𝐽 fLim 𝑔))
13	1, 12	sselid 3933	. . . 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 3130	. 2 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → ∀𝑔 ∈ (Fil‘𝑋)(𝐹 ⊆ 𝑔 → 𝐴 ∈ (𝐽 fClus 𝑔)))
16		sseq2 3962	. . . . . 6 ⊢ (𝑔 = 𝐹 → (𝐹 ⊆ 𝑔 ↔ 𝐹 ⊆ 𝐹))
17		oveq2 7376	. . . . . . 7 ⊢ (𝑔 = 𝐹 → (𝐽 fClus 𝑔) = (𝐽 fClus 𝐹))
18	17	eleq2d 2823	. . . . . 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 3574	. . . 4 ⊢ (𝐹 ∈ (Fil‘𝑋) → (∀𝑔 ∈ (Fil‘𝑋)(𝐹 ⊆ 𝑔 → 𝐴 ∈ (𝐽 fClus 𝑔)) → (𝐹 ⊆ 𝐹 → 𝐴 ∈ (𝐽 fClus 𝐹))))
21		ssid 3958	. . . . . 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 23967	. . . . . 6 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐽 ∈ Top)
25	3	fclselbas 23972	. . . . . 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 4426	. . . . . . . . . . . . . 14 ⊢ (𝑜 ∩ (𝑋 ∖ 𝑜)) = ∅
30		simpll 767	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → 𝐹 ∈ (Fil‘𝑋))
31		simplrl 777	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → 𝐽 ∈ Top)
32	3	topopn 22862	. . . . . . . . . . . . . . . . . . . . . . 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 5325	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑋 ∈ 𝐽 → 𝒫 𝑋 ∈ V)
35		rabexg 5284	. . . . . . . . . . . . . . . . . . . . . 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 7698	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥} ∈ V) → (𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥}) ∈ V)
38	30, 36, 37	syl2anc 585	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → (𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥}) ∈ V)
39		ssfii 9334	. . . . . . . . . . . . . . . . . . . 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 23807	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ⊆ 𝒫 𝑋)
42		ssrab2 4034	. . . . . . . . . . . . . . . . . . . . . . . 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 4146	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥}) ⊆ 𝒫 𝑋)
45	44	ad2antrr 727	. . . . . . . . . . . . . . . . . . . . 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 4133	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥} ⊆ (𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥})
47		sseq2 3962	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 = (𝑋 ∖ 𝑜) → ((𝑋 ∖ 𝑜) ⊆ 𝑥 ↔ (𝑋 ∖ 𝑜) ⊆ (𝑋 ∖ 𝑜)))
48		difss 4090	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑋 ∖ 𝑜) ⊆ 𝑋
49		elpw2g 5280	. . . . . . . . . . . . . . . . . . . . . . . . . 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 3958	. . . . . . . . . . . . . . . . . . . . . . . . 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 3650	. . . . . . . . . . . . . . . . . . . . . . 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 3933	. . . . . . . . . . . . . . . . . . . . . 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 4296	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → (𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥}) ≠ ∅)
57		sseq2 3962	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑥 = 𝑧 → ((𝑋 ∖ 𝑜) ⊆ 𝑥 ↔ (𝑋 ∖ 𝑜) ⊆ 𝑧))
58	57	elrab 3648	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑧 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥} ↔ (𝑧 ∈ 𝒫 𝑋 ∧ (𝑋 ∖ 𝑜) ⊆ 𝑧))
59	58	simprbi 497	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑧 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥} → (𝑋 ∖ 𝑜) ⊆ 𝑧)
60	59	ad2antll 730	. . . . . . . . . . . . . . . . . . . . . . . . 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 4197	. . . . . . . . . . . . . . . . . . . . . . . . 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 782	. . . . . . . . . . . . . . . . . . . . . . . . . 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 4328	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑦 ∩ 𝑋) ⊆ 𝑜 ↔ (𝑦 ∩ (𝑋 ∖ 𝑜)) = ∅)
66		simplll 775	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 771	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 23808	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑦 ∈ 𝐹) → 𝑦 ⊆ 𝑋)
69	66, 67, 68	syl2anc 585	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3921	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3967	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 727	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 777	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4896	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑜 ∈ 𝐽 → 𝑜 ⊆ ∪ 𝐽)
76	75, 3	sseqtrrdi 3977	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑜 ∈ 𝐽 → 𝑜 ⊆ 𝑋)
77	76	ad2antrl 729	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → 𝑜 ⊆ 𝑋)
78	77	ad2antrr 727	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 23809	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑦 ∈ 𝐹 ∧ 𝑜 ⊆ 𝑋 ∧ 𝑦 ⊆ 𝑜)) → 𝑜 ∈ 𝐹)
81	73, 74, 78, 79, 80	syl13anc 1375	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2947	. . . . . . . . . . . . . . . . . . . . . . . . 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 4358	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑦 ∩ (𝑋 ∖ 𝑜)) ⊆ (𝑦 ∩ 𝑧) ∧ (𝑦 ∩ (𝑋 ∖ 𝑜)) ≠ ∅) → (𝑦 ∩ 𝑧) ≠ ∅)
88	62, 86, 87	syl2anc 585	. . . . . . . . . . . . . . . . . . . . . . 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 3181	. . . . . . . . . . . . . . . . . . . . . 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 23804	. . . . . . . . . . . . . . . . . . . . . . . 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 23811	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2825	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2947	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 4322	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑜 ⊆ 𝑋 ∧ 𝑜 ≠ 𝑋) → (𝑋 ∖ 𝑜) ≠ ∅)
100	77, 98, 99	syl2anc 585	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → (𝑋 ∖ 𝑜) ≠ ∅)
101		supfil 23851	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑋 ∈ 𝐽 ∧ (𝑋 ∖ 𝑜) ⊆ 𝑋 ∧ (𝑋 ∖ 𝑜) ≠ ∅) → {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥} ∈ (Fil‘𝑋))
102	33, 92, 100, 101	syl3anc 1374	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥} ∈ (Fil‘𝑋))
103		filfbas 23804	. . . . . . . . . . . . . . . . . . . . . . . 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 23825	. . . . . . . . . . . . . . . . . . . . . . 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 585	. . . . . . . . . . . . . . . . . . . . . 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 23823	. . . . . . . . . . . . . . . . . . . . . 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 1344	. . . . . . . . . . . . . . . . . . . 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 23828	. . . . . . . . . . . . . . . . . . . 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 3946	. . . . . . . . . . . . . . . . . 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 4147	. . . . . . . . . . . . . . . . 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 23834	. . . . . . . . . . . . . . . . . . 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 3962	. . . . . . . . . . . . . . . . . . . 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 7376	. . . . . . . . . . . . . . . . . . . . 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 2823	. . . . . . . . . . . . . . . . . . . 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 3574	. . . . . . . . . . . . . . . . . 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 777	. . . . . . . . . . . . . . . . . 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 781	. . . . . . . . . . . . . . . . . . 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 3936	. . . . . . . . . . . . . . . . . . 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 23971	. . . . . . . . . . . . . . . . . 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 1375	. . . . . . . . . . . . . . . . 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 2949	. . . . . . . . . . . . . 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 3138	. . . . . . 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 773	. . . . . . 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 771	. . . . . . . 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 23931	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fLim 𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → 𝑜 ∈ 𝐹))))
148	145, 146, 147	syl2anc 585	. . . . . 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 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  {crab 3401  Vcvv 3442   ∖ cdif 3900   ∪ cun 3901   ∩ cin 3902   ⊆ wss 3903  ∅c0 4287  𝒫 cpw 4556  ∪ cuni 4865  ‘cfv 6500  (class class class)co 7368  ficfi 9325  fBascfbas 21309  filGencfg 21310  Topctop 22849  TopOnctopon 22866  Filcfil 23801   fLim cflim 23890   fClus cfcls 23892

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 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-iin 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-1o 8407  df-2o 8408  df-en 8896  df-fin 8899  df-fi 9326  df-fbas 21318  df-fg 21319  df-top 22850  df-topon 22867  df-cld 22975  df-ntr 22976  df-cls 22977  df-nei 23054  df-fil 23802  df-flim 23895  df-fcls 23897

This theorem is referenced by:  cnpfcf  23997