Theorem fclsfnflim 24084

Description: A filter clusters at a point iff a finer filter converges to it. (Contributed by Jeff Hankins, 12-Nov-2009.) (Revised by Mario Carneiro, 26-Aug-2015.)

Assertion

Ref	Expression
fclsfnflim	⊢ (𝐹 ∈ (Fil‘𝑋) → (𝐴 ∈ (𝐽 fClus 𝐹) ↔ ∃𝑔 ∈ (Fil‘𝑋)(𝐹 ⊆ 𝑔 ∧ 𝐴 ∈ (𝐽 fLim 𝑔))))

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

Proof of Theorem fclsfnflim

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

Step	Hyp	Ref	Expression
1		filsspw 23908	. . . . . . . 8 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ⊆ 𝒫 𝑋)
2	1	adantr 484	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → 𝐹 ⊆ 𝒫 𝑋)
3		fclstop 24068	. . . . . . . . . 10 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐽 ∈ Top)
4	3	adantl 485	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → 𝐽 ∈ Top)
5		eqid 2762	. . . . . . . . . 10 ⊢ ∪ 𝐽 = ∪ 𝐽
6	5	neisspw 23164	. . . . . . . . 9 ⊢ (𝐽 ∈ Top → ((nei‘𝐽)‘{𝐴}) ⊆ 𝒫 ∪ 𝐽)
7	4, 6	syl 17	. . . . . . . 8 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → ((nei‘𝐽)‘{𝐴}) ⊆ 𝒫 ∪ 𝐽)
8		filunibas 23938	. . . . . . . . . 10 ⊢ (𝐹 ∈ (Fil‘𝑋) → ∪ 𝐹 = 𝑋)
9	5	fclsfil 24067	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐹 ∈ (Fil‘∪ 𝐽))
10		filunibas 23938	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (Fil‘∪ 𝐽) → ∪ 𝐹 = ∪ 𝐽)
11	9, 10	syl 17	. . . . . . . . . 10 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → ∪ 𝐹 = ∪ 𝐽)
12	8, 11	sylan9req 2818	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → 𝑋 = ∪ 𝐽)
13	12	pweqd 4572	. . . . . . . 8 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → 𝒫 𝑋 = 𝒫 ∪ 𝐽)
14	7, 13	sseqtrrd 3973	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → ((nei‘𝐽)‘{𝐴}) ⊆ 𝒫 𝑋)
15	2, 14	unssd 4144	. . . . . 6 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → (𝐹 ∪ ((nei‘𝐽)‘{𝐴})) ⊆ 𝒫 𝑋)
16		ssun1 4130	. . . . . . . 8 ⊢ 𝐹 ⊆ (𝐹 ∪ ((nei‘𝐽)‘{𝐴}))
17		filn0 23919	. . . . . . . 8 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ≠ ∅)
18		ssn0 4358	. . . . . . . 8 ⊢ ((𝐹 ⊆ (𝐹 ∪ ((nei‘𝐽)‘{𝐴})) ∧ 𝐹 ≠ ∅) → (𝐹 ∪ ((nei‘𝐽)‘{𝐴})) ≠ ∅)
19	16, 17, 18	sylancr 596	. . . . . . 7 ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝐹 ∪ ((nei‘𝐽)‘{𝐴})) ≠ ∅)
20	19	adantr 484	. . . . . 6 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → (𝐹 ∪ ((nei‘𝐽)‘{𝐴})) ≠ ∅)
21		incom 4161	. . . . . . . . . . . 12 ⊢ (𝑦 ∩ 𝑥) = (𝑥 ∩ 𝑦)
22		fclsneii 24074	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑥 ∈ 𝐹) → (𝑦 ∩ 𝑥) ≠ ∅)
23	21, 22	eqnetrrid 3032	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑥 ∈ 𝐹) → (𝑥 ∩ 𝑦) ≠ ∅)
24	23	3com23 1139	. . . . . . . . . 10 ⊢ ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑥 ∈ 𝐹 ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → (𝑥 ∩ 𝑦) ≠ ∅)
25	24	3expb 1133	. . . . . . . . 9 ⊢ ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴}))) → (𝑥 ∩ 𝑦) ≠ ∅)
26	25	adantll 724	. . . . . . . 8 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴}))) → (𝑥 ∩ 𝑦) ≠ ∅)
27	26	ralrimivva 3205	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ ((nei‘𝐽)‘{𝐴})(𝑥 ∩ 𝑦) ≠ ∅)
28		filfbas 23905	. . . . . . . . 9 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋))
29	28	adantr 484	. . . . . . . 8 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → 𝐹 ∈ (fBas‘𝑋))
30		istopon 22969	. . . . . . . . . . 11 ⊢ (𝐽 ∈ (TopOn‘𝑋) ↔ (𝐽 ∈ Top ∧ 𝑋 = ∪ 𝐽))
31	4, 12, 30	sylanbrc 592	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → 𝐽 ∈ (TopOn‘𝑋))
32	5	fclselbas 24073	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐴 ∈ ∪ 𝐽)
33	32	adantl 485	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → 𝐴 ∈ ∪ 𝐽)
34	33, 12	eleqtrrd 2865	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → 𝐴 ∈ 𝑋)
35	34	snssd 4745	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → {𝐴} ⊆ 𝑋)
36		snnzg 4733	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → {𝐴} ≠ ∅)
37	36	adantl 485	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → {𝐴} ≠ ∅)
38		neifil 23937	. . . . . . . . . 10 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ {𝐴} ⊆ 𝑋 ∧ {𝐴} ≠ ∅) → ((nei‘𝐽)‘{𝐴}) ∈ (Fil‘𝑋))
39	31, 35, 37, 38	syl3anc 1390	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → ((nei‘𝐽)‘{𝐴}) ∈ (Fil‘𝑋))
40		filfbas 23905	. . . . . . . . 9 ⊢ (((nei‘𝐽)‘{𝐴}) ∈ (Fil‘𝑋) → ((nei‘𝐽)‘{𝐴}) ∈ (fBas‘𝑋))
41	39, 40	syl 17	. . . . . . . 8 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → ((nei‘𝐽)‘{𝐴}) ∈ (fBas‘𝑋))
42		fbunfip 23926	. . . . . . . 8 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ ((nei‘𝐽)‘{𝐴}) ∈ (fBas‘𝑋)) → (¬ ∅ ∈ (fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴}))) ↔ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ ((nei‘𝐽)‘{𝐴})(𝑥 ∩ 𝑦) ≠ ∅))
43	29, 41, 42	syl2anc 593	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → (¬ ∅ ∈ (fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴}))) ↔ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ ((nei‘𝐽)‘{𝐴})(𝑥 ∩ 𝑦) ≠ ∅))
44	27, 43	mpbird 259	. . . . . 6 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → ¬ ∅ ∈ (fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴}))))
45		filtop 23912	. . . . . . . 8 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝑋 ∈ 𝐹)
46		fsubbas 23924	. . . . . . . 8 ⊢ (𝑋 ∈ 𝐹 → ((fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴}))) ∈ (fBas‘𝑋) ↔ ((𝐹 ∪ ((nei‘𝐽)‘{𝐴})) ⊆ 𝒫 𝑋 ∧ (𝐹 ∪ ((nei‘𝐽)‘{𝐴})) ≠ ∅ ∧ ¬ ∅ ∈ (fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴}))))))
47	45, 46	syl 17	. . . . . . 7 ⊢ (𝐹 ∈ (Fil‘𝑋) → ((fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴}))) ∈ (fBas‘𝑋) ↔ ((𝐹 ∪ ((nei‘𝐽)‘{𝐴})) ⊆ 𝒫 𝑋 ∧ (𝐹 ∪ ((nei‘𝐽)‘{𝐴})) ≠ ∅ ∧ ¬ ∅ ∈ (fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴}))))))
48	47	adantr 484	. . . . . 6 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → ((fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴}))) ∈ (fBas‘𝑋) ↔ ((𝐹 ∪ ((nei‘𝐽)‘{𝐴})) ⊆ 𝒫 𝑋 ∧ (𝐹 ∪ ((nei‘𝐽)‘{𝐴})) ≠ ∅ ∧ ¬ ∅ ∈ (fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴}))))))
49	15, 20, 44, 48	mpbir3and 1356	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → (fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴}))) ∈ (fBas‘𝑋))
50		fgcl 23935	. . . . 5 ⊢ ((fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴}))) ∈ (fBas‘𝑋) → (𝑋filGen(fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴})))) ∈ (Fil‘𝑋))
51	49, 50	syl 17	. . . 4 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → (𝑋filGen(fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴})))) ∈ (Fil‘𝑋))
52		fvex 6880	. . . . . . . . 9 ⊢ ((nei‘𝐽)‘{𝐴}) ∈ V
53		unexg 7726	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ ((nei‘𝐽)‘{𝐴}) ∈ V) → (𝐹 ∪ ((nei‘𝐽)‘{𝐴})) ∈ V)
54	52, 53	mpan2 701	. . . . . . . 8 ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝐹 ∪ ((nei‘𝐽)‘{𝐴})) ∈ V)
55		ssfii 9365	. . . . . . . 8 ⊢ ((𝐹 ∪ ((nei‘𝐽)‘{𝐴})) ∈ V → (𝐹 ∪ ((nei‘𝐽)‘{𝐴})) ⊆ (fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴}))))
56	54, 55	syl 17	. . . . . . 7 ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝐹 ∪ ((nei‘𝐽)‘{𝐴})) ⊆ (fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴}))))
57	56	adantr 484	. . . . . 6 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → (𝐹 ∪ ((nei‘𝐽)‘{𝐴})) ⊆ (fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴}))))
58	57	unssad 4145	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → 𝐹 ⊆ (fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴}))))
59		ssfg 23929	. . . . . 6 ⊢ ((fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴}))) ∈ (fBas‘𝑋) → (fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴}))) ⊆ (𝑋filGen(fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴})))))
60	49, 59	syl 17	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → (fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴}))) ⊆ (𝑋filGen(fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴})))))
61	58, 60	sstrd 3946	. . . 4 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → 𝐹 ⊆ (𝑋filGen(fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴})))))
62	57	unssbd 4146	. . . . . 6 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → ((nei‘𝐽)‘{𝐴}) ⊆ (fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴}))))
63	62, 60	sstrd 3946	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → ((nei‘𝐽)‘{𝐴}) ⊆ (𝑋filGen(fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴})))))
64		elflim 24028	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑋filGen(fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴})))) ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fLim (𝑋filGen(fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴}))))) ↔ (𝐴 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ (𝑋filGen(fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴})))))))
65	31, 51, 64	syl2anc 593	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → (𝐴 ∈ (𝐽 fLim (𝑋filGen(fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴}))))) ↔ (𝐴 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ (𝑋filGen(fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴})))))))
66	34, 63, 65	mpbir2and 723	. . . 4 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → 𝐴 ∈ (𝐽 fLim (𝑋filGen(fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴}))))))
67		sseq2 3962	. . . . . 6 ⊢ (𝑔 = (𝑋filGen(fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴})))) → (𝐹 ⊆ 𝑔 ↔ 𝐹 ⊆ (𝑋filGen(fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴}))))))
68		oveq2 7404	. . . . . . 7 ⊢ (𝑔 = (𝑋filGen(fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴})))) → (𝐽 fLim 𝑔) = (𝐽 fLim (𝑋filGen(fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴}))))))
69	68	eleq2d 2848	. . . . . 6 ⊢ (𝑔 = (𝑋filGen(fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴})))) → (𝐴 ∈ (𝐽 fLim 𝑔) ↔ 𝐴 ∈ (𝐽 fLim (𝑋filGen(fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴})))))))
70	67, 69	anbi12d 641	. . . . 5 ⊢ (𝑔 = (𝑋filGen(fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴})))) → ((𝐹 ⊆ 𝑔 ∧ 𝐴 ∈ (𝐽 fLim 𝑔)) ↔ (𝐹 ⊆ (𝑋filGen(fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴})))) ∧ 𝐴 ∈ (𝐽 fLim (𝑋filGen(fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴}))))))))
71	70	rspcev 3581	. . . 4 ⊢ (((𝑋filGen(fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴})))) ∈ (Fil‘𝑋) ∧ (𝐹 ⊆ (𝑋filGen(fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴})))) ∧ 𝐴 ∈ (𝐽 fLim (𝑋filGen(fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴}))))))) → ∃𝑔 ∈ (Fil‘𝑋)(𝐹 ⊆ 𝑔 ∧ 𝐴 ∈ (𝐽 fLim 𝑔)))
72	51, 61, 66, 71	syl12anc 847	. . 3 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → ∃𝑔 ∈ (Fil‘𝑋)(𝐹 ⊆ 𝑔 ∧ 𝐴 ∈ (𝐽 fLim 𝑔)))
73	72	ex 416	. 2 ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝐴 ∈ (𝐽 fClus 𝐹) → ∃𝑔 ∈ (Fil‘𝑋)(𝐹 ⊆ 𝑔 ∧ 𝐴 ∈ (𝐽 fLim 𝑔))))
74		simprl 780	. . . . . 6 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ (𝐹 ⊆ 𝑔 ∧ 𝐴 ∈ (𝐽 fLim 𝑔)))) → 𝑔 ∈ (Fil‘𝑋))
75		simprrr 791	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ (𝐹 ⊆ 𝑔 ∧ 𝐴 ∈ (𝐽 fLim 𝑔)))) → 𝐴 ∈ (𝐽 fLim 𝑔))
76		flimtopon 24027	. . . . . . 7 ⊢ (𝐴 ∈ (𝐽 fLim 𝑔) → (𝐽 ∈ (TopOn‘𝑋) ↔ 𝑔 ∈ (Fil‘𝑋)))
77	75, 76	syl 17	. . . . . 6 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ (𝐹 ⊆ 𝑔 ∧ 𝐴 ∈ (𝐽 fLim 𝑔)))) → (𝐽 ∈ (TopOn‘𝑋) ↔ 𝑔 ∈ (Fil‘𝑋)))
78	74, 77	mpbird 259	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ (𝐹 ⊆ 𝑔 ∧ 𝐴 ∈ (𝐽 fLim 𝑔)))) → 𝐽 ∈ (TopOn‘𝑋))
79		simpl 486	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ (𝐹 ⊆ 𝑔 ∧ 𝐴 ∈ (𝐽 fLim 𝑔)))) → 𝐹 ∈ (Fil‘𝑋))
80		simprrl 790	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ (𝐹 ⊆ 𝑔 ∧ 𝐴 ∈ (𝐽 fLim 𝑔)))) → 𝐹 ⊆ 𝑔)
81		fclsss2 24080	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝑔) → (𝐽 fClus 𝑔) ⊆ (𝐽 fClus 𝐹))
82	78, 79, 80, 81	syl3anc 1390	. . . 4 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ (𝐹 ⊆ 𝑔 ∧ 𝐴 ∈ (𝐽 fLim 𝑔)))) → (𝐽 fClus 𝑔) ⊆ (𝐽 fClus 𝐹))
83		flimfcls 24083	. . . . 5 ⊢ (𝐽 fLim 𝑔) ⊆ (𝐽 fClus 𝑔)
84	83, 75	sselid 3934	. . . 4 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ (𝐹 ⊆ 𝑔 ∧ 𝐴 ∈ (𝐽 fLim 𝑔)))) → 𝐴 ∈ (𝐽 fClus 𝑔))
85	82, 84	sseldd 3937	. . 3 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ (𝐹 ⊆ 𝑔 ∧ 𝐴 ∈ (𝐽 fLim 𝑔)))) → 𝐴 ∈ (𝐽 fClus 𝐹))
86	85	rexlimdvaa 3164	. 2 ⊢ (𝐹 ∈ (Fil‘𝑋) → (∃𝑔 ∈ (Fil‘𝑋)(𝐹 ⊆ 𝑔 ∧ 𝐴 ∈ (𝐽 fLim 𝑔)) → 𝐴 ∈ (𝐽 fClus 𝐹)))
87	73, 86	impbid 214	1 ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝐴 ∈ (𝐽 fClus 𝐹) ↔ ∃𝑔 ∈ (Fil‘𝑋)(𝐹 ⊆ 𝑔 ∧ 𝐴 ∈ (𝐽 fLim 𝑔))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1098   = wceq 1560   ∈ wcel 2142   ≠ wne 2957  ∀wral 3076  ∃wrex 3086  Vcvv 3454   ∪ cun 3902   ∩ cin 3903   ⊆ wss 3904  ∅c0 4285  𝒫 cpw 4555  {csn 4582  ∪ cuni 4865  ‘cfv 6521  (class class class)co 7396  ficfi 9356  fBascfbas 21409  filGencfg 21410  Topctop 22950  TopOnctopon 22967  neicnei 23154  Filcfil 23902   fLim cflim 23991   fClus cfcls 23993

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3062  df-ral 3077  df-rex 3087  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4906  df-iun 4951  df-iin 4952  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-ov 7399  df-oprab 7400  df-mpo 7401  df-om 7847  df-1o 8437  df-2o 8438  df-en 8928  df-fin 8931  df-fi 9357  df-fbas 21418  df-fg 21419  df-top 22951  df-topon 22968  df-cld 23076  df-ntr 23077  df-cls 23078  df-nei 23155  df-fil 23903  df-flim 23996  df-fcls 23998

This theorem is referenced by:  uffclsflim  24088  cnpfcfi  24097