Theorem flimfnfcls 24003

Description: A filter converges to a point iff every finer filter clusters there. Along with fclsfnflim 24002, 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 24001	. . . . 5 ⊢ (𝐽 fLim 𝑔) ⊆ (𝐽 fClus 𝑔)
2		flimtop 23940	. . . . . . . . 9 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐽 ∈ Top)
3		flimfnfcls.x	. . . . . . . . . 10 ⊢ 𝑋 = ∪ 𝐽
4	3	toptopon 22892	. . . . . . . . 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 23947	. . . . . . 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 3923	. . . . 5 ⊢ (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑔 ∈ (Fil‘𝑋)) ∧ 𝐹 ⊆ 𝑔) → 𝐴 ∈ (𝐽 fLim 𝑔))
13	1, 12	sselid 3920	. . . 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 3949	. . . . . 6 ⊢ (𝑔 = 𝐹 → (𝐹 ⊆ 𝑔 ↔ 𝐹 ⊆ 𝐹))
17		oveq2 7368	. . . . . . 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 3561	. . . 4 ⊢ (𝐹 ∈ (Fil‘𝑋) → (∀𝑔 ∈ (Fil‘𝑋)(𝐹 ⊆ 𝑔 → 𝐴 ∈ (𝐽 fClus 𝑔)) → (𝐹 ⊆ 𝐹 → 𝐴 ∈ (𝐽 fClus 𝐹))))
21		ssid 3945	. . . . . 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 23986	. . . . . 6 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐽 ∈ Top)
25	3	fclselbas 23991	. . . . . 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 4413	. . . . . . . . . . . . . 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 22881	. . . . . . . . . . . . . . . . . . . . . . 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 5315	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑋 ∈ 𝐽 → 𝒫 𝑋 ∈ V)
35		rabexg 5274	. . . . . . . . . . . . . . . . . . . . . 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 7690	. . . . . . . . . . . . . . . . . . . . 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 9325	. . . . . . . . . . . . . . . . . . . 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 23826	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ⊆ 𝒫 𝑋)
42		ssrab2 4021	. . . . . . . . . . . . . . . . . . . . . . . 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 4133	. . . . . . . . . . . . . . . . . . . . . 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 4120	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥} ⊆ (𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥})
47		sseq2 3949	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 = (𝑋 ∖ 𝑜) → ((𝑋 ∖ 𝑜) ⊆ 𝑥 ↔ (𝑋 ∖ 𝑜) ⊆ (𝑋 ∖ 𝑜)))
48		difss 4077	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑋 ∖ 𝑜) ⊆ 𝑋
49		elpw2g 5270	. . . . . . . . . . . . . . . . . . . . . . . . . 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 3945	. . . . . . . . . . . . . . . . . . . . . . . . 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 3637	. . . . . . . . . . . . . . . . . . . . . . 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 3920	. . . . . . . . . . . . . . . . . . . . . 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 4283	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → (𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥}) ≠ ∅)
57		sseq2 3949	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑥 = 𝑧 → ((𝑋 ∖ 𝑜) ⊆ 𝑥 ↔ (𝑋 ∖ 𝑜) ⊆ 𝑧))
58	57	elrab 3635	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 4184	. . . . . . . . . . . . . . . . . . . . . . . . 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 4315	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 23827	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3908	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3954	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4882	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑜 ∈ 𝐽 → 𝑜 ⊆ ∪ 𝐽)
76	75, 3	sseqtrrdi 3964	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 23828	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4345	. . . . . . . . . . . . . . . . . . . . . . . 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 23823	. . . . . . . . . . . . . . . . . . . . . . . 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 23830	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4309	. . . . . . . . . . . . . . . . . . . . . . . . . 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 23870	. . . . . . . . . . . . . . . . . . . . . . . . 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 23823	. . . . . . . . . . . . . . . . . . . . . . . 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 23844	. . . . . . . . . . . . . . . . . . . . . . 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 23842	. . . . . . . . . . . . . . . . . . . . . 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 23847	. . . . . . . . . . . . . . . . . . . 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 3933	. . . . . . . . . . . . . . . . . 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 4134	. . . . . . . . . . . . . . . . 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 23853	. . . . . . . . . . . . . . . . . . 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 3949	. . . . . . . . . . . . . . . . . . . 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 7368	. . . . . . . . . . . . . . . . . . . . 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 3561	. . . . . . . . . . . . . . . . . 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 3923	. . . . . . . . . . . . . . . . . . 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 23990	. . . . . . . . . . . . . . . . . 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 23950	. . . . . . 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 3390  Vcvv 3430   ∖ cdif 3887   ∪ cun 3888   ∩ cin 3889   ⊆ wss 3890  ∅c0 4274  𝒫 cpw 4542  ∪ cuni 4851  ‘cfv 6492  (class class class)co 7360  ficfi 9316  fBascfbas 21332  filGencfg 21333  Topctop 22868  TopOnctopon 22885  Filcfil 23820   fLim cflim 23909   fClus cfcls 23911

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 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

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 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-iin 4937  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1o 8398  df-2o 8399  df-en 8887  df-fin 8890  df-fi 9317  df-fbas 21341  df-fg 21342  df-top 22869  df-topon 22886  df-cld 22994  df-ntr 22995  df-cls 22996  df-nei 23073  df-fil 23821  df-flim 23914  df-fcls 23916

This theorem is referenced by:  cnpfcf  24016