Theorem fclsfnflim 24036

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 23860	. . . . . . . 8 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ⊆ 𝒫 𝑋)
2	1	adantr 480	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → 𝐹 ⊆ 𝒫 𝑋)
3		fclstop 24020	. . . . . . . . . 10 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐽 ∈ Top)
4	3	adantl 481	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → 𝐽 ∈ Top)
5		eqid 2736	. . . . . . . . . 10 ⊢ ∪ 𝐽 = ∪ 𝐽
6	5	neisspw 23116	. . . . . . . . 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 23890	. . . . . . . . . 10 ⊢ (𝐹 ∈ (Fil‘𝑋) → ∪ 𝐹 = 𝑋)
9	5	fclsfil 24019	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐹 ∈ (Fil‘∪ 𝐽))
10		filunibas 23890	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (Fil‘∪ 𝐽) → ∪ 𝐹 = ∪ 𝐽)
11	9, 10	syl 17	. . . . . . . . . 10 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → ∪ 𝐹 = ∪ 𝐽)
12	8, 11	sylan9req 2797	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → 𝑋 = ∪ 𝐽)
13	12	pweqd 4616	. . . . . . . 8 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → 𝒫 𝑋 = 𝒫 ∪ 𝐽)
14	7, 13	sseqtrrd 4020	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → ((nei‘𝐽)‘{𝐴}) ⊆ 𝒫 𝑋)
15	2, 14	unssd 4191	. . . . . 6 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → (𝐹 ∪ ((nei‘𝐽)‘{𝐴})) ⊆ 𝒫 𝑋)
16		ssun1 4177	. . . . . . . 8 ⊢ 𝐹 ⊆ (𝐹 ∪ ((nei‘𝐽)‘{𝐴}))
17		filn0 23871	. . . . . . . 8 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ≠ ∅)
18		ssn0 4403	. . . . . . . 8 ⊢ ((𝐹 ⊆ (𝐹 ∪ ((nei‘𝐽)‘{𝐴})) ∧ 𝐹 ≠ ∅) → (𝐹 ∪ ((nei‘𝐽)‘{𝐴})) ≠ ∅)
19	16, 17, 18	sylancr 587	. . . . . . 7 ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝐹 ∪ ((nei‘𝐽)‘{𝐴})) ≠ ∅)
20	19	adantr 480	. . . . . 6 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → (𝐹 ∪ ((nei‘𝐽)‘{𝐴})) ≠ ∅)
21		incom 4208	. . . . . . . . . . . 12 ⊢ (𝑦 ∩ 𝑥) = (𝑥 ∩ 𝑦)
22		fclsneii 24026	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑥 ∈ 𝐹) → (𝑦 ∩ 𝑥) ≠ ∅)
23	21, 22	eqnetrrid 3015	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑥 ∈ 𝐹) → (𝑥 ∩ 𝑦) ≠ ∅)
24	23	3com23 1126	. . . . . . . . . 10 ⊢ ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑥 ∈ 𝐹 ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → (𝑥 ∩ 𝑦) ≠ ∅)
25	24	3expb 1120	. . . . . . . . 9 ⊢ ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴}))) → (𝑥 ∩ 𝑦) ≠ ∅)
26	25	adantll 714	. . . . . . . 8 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴}))) → (𝑥 ∩ 𝑦) ≠ ∅)
27	26	ralrimivva 3201	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ ((nei‘𝐽)‘{𝐴})(𝑥 ∩ 𝑦) ≠ ∅)
28		filfbas 23857	. . . . . . . . 9 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋))
29	28	adantr 480	. . . . . . . 8 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → 𝐹 ∈ (fBas‘𝑋))
30		istopon 22919	. . . . . . . . . . 11 ⊢ (𝐽 ∈ (TopOn‘𝑋) ↔ (𝐽 ∈ Top ∧ 𝑋 = ∪ 𝐽))
31	4, 12, 30	sylanbrc 583	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → 𝐽 ∈ (TopOn‘𝑋))
32	5	fclselbas 24025	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐴 ∈ ∪ 𝐽)
33	32	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → 𝐴 ∈ ∪ 𝐽)
34	33, 12	eleqtrrd 2843	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → 𝐴 ∈ 𝑋)
35	34	snssd 4808	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → {𝐴} ⊆ 𝑋)
36		snnzg 4773	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → {𝐴} ≠ ∅)
37	36	adantl 481	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → {𝐴} ≠ ∅)
38		neifil 23889	. . . . . . . . . 10 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ {𝐴} ⊆ 𝑋 ∧ {𝐴} ≠ ∅) → ((nei‘𝐽)‘{𝐴}) ∈ (Fil‘𝑋))
39	31, 35, 37, 38	syl3anc 1372	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → ((nei‘𝐽)‘{𝐴}) ∈ (Fil‘𝑋))
40		filfbas 23857	. . . . . . . . 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 23878	. . . . . . . 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 584	. . . . . . 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 257	. . . . . 6 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → ¬ ∅ ∈ (fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴}))))
45		filtop 23864	. . . . . . . 8 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝑋 ∈ 𝐹)
46		fsubbas 23876	. . . . . . . 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 480	. . . . . 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 1342	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → (fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴}))) ∈ (fBas‘𝑋))
50		fgcl 23887	. . . . 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 6918	. . . . . . . . 9 ⊢ ((nei‘𝐽)‘{𝐴}) ∈ V
53		unexg 7764	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ ((nei‘𝐽)‘{𝐴}) ∈ V) → (𝐹 ∪ ((nei‘𝐽)‘{𝐴})) ∈ V)
54	52, 53	mpan2 691	. . . . . . . 8 ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝐹 ∪ ((nei‘𝐽)‘{𝐴})) ∈ V)
55		ssfii 9460	. . . . . . . 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 480	. . . . . 6 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → (𝐹 ∪ ((nei‘𝐽)‘{𝐴})) ⊆ (fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴}))))
58	57	unssad 4192	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → 𝐹 ⊆ (fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴}))))
59		ssfg 23881	. . . . . 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 3993	. . . 4 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → 𝐹 ⊆ (𝑋filGen(fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴})))))
62	57	unssbd 4193	. . . . . 6 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → ((nei‘𝐽)‘{𝐴}) ⊆ (fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴}))))
63	62, 60	sstrd 3993	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → ((nei‘𝐽)‘{𝐴}) ⊆ (𝑋filGen(fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴})))))
64		elflim 23980	. . . . . 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 584	. . . . 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 713	. . . 4 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → 𝐴 ∈ (𝐽 fLim (𝑋filGen(fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴}))))))
67		sseq2 4009	. . . . . 6 ⊢ (𝑔 = (𝑋filGen(fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴})))) → (𝐹 ⊆ 𝑔 ↔ 𝐹 ⊆ (𝑋filGen(fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴}))))))
68		oveq2 7440	. . . . . . 7 ⊢ (𝑔 = (𝑋filGen(fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴})))) → (𝐽 fLim 𝑔) = (𝐽 fLim (𝑋filGen(fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴}))))))
69	68	eleq2d 2826	. . . . . 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 632	. . . . 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 3621	. . . 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 836	. . 3 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → ∃𝑔 ∈ (Fil‘𝑋)(𝐹 ⊆ 𝑔 ∧ 𝐴 ∈ (𝐽 fLim 𝑔)))
73	72	ex 412	. 2 ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝐴 ∈ (𝐽 fClus 𝐹) → ∃𝑔 ∈ (Fil‘𝑋)(𝐹 ⊆ 𝑔 ∧ 𝐴 ∈ (𝐽 fLim 𝑔))))
74		simprl 770	. . . . . 6 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ (𝐹 ⊆ 𝑔 ∧ 𝐴 ∈ (𝐽 fLim 𝑔)))) → 𝑔 ∈ (Fil‘𝑋))
75		simprrr 781	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ (𝐹 ⊆ 𝑔 ∧ 𝐴 ∈ (𝐽 fLim 𝑔)))) → 𝐴 ∈ (𝐽 fLim 𝑔))
76		flimtopon 23979	. . . . . . 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 257	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ (𝐹 ⊆ 𝑔 ∧ 𝐴 ∈ (𝐽 fLim 𝑔)))) → 𝐽 ∈ (TopOn‘𝑋))
79		simpl 482	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ (𝐹 ⊆ 𝑔 ∧ 𝐴 ∈ (𝐽 fLim 𝑔)))) → 𝐹 ∈ (Fil‘𝑋))
80		simprrl 780	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ (𝐹 ⊆ 𝑔 ∧ 𝐴 ∈ (𝐽 fLim 𝑔)))) → 𝐹 ⊆ 𝑔)
81		fclsss2 24032	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝑔) → (𝐽 fClus 𝑔) ⊆ (𝐽 fClus 𝐹))
82	78, 79, 80, 81	syl3anc 1372	. . . 4 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ (𝐹 ⊆ 𝑔 ∧ 𝐴 ∈ (𝐽 fLim 𝑔)))) → (𝐽 fClus 𝑔) ⊆ (𝐽 fClus 𝐹))
83		flimfcls 24035	. . . . 5 ⊢ (𝐽 fLim 𝑔) ⊆ (𝐽 fClus 𝑔)
84	83, 75	sselid 3980	. . . 4 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ (𝐹 ⊆ 𝑔 ∧ 𝐴 ∈ (𝐽 fLim 𝑔)))) → 𝐴 ∈ (𝐽 fClus 𝑔))
85	82, 84	sseldd 3983	. . 3 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ (𝐹 ⊆ 𝑔 ∧ 𝐴 ∈ (𝐽 fLim 𝑔)))) → 𝐴 ∈ (𝐽 fClus 𝐹))
86	85	rexlimdvaa 3155	. 2 ⊢ (𝐹 ∈ (Fil‘𝑋) → (∃𝑔 ∈ (Fil‘𝑋)(𝐹 ⊆ 𝑔 ∧ 𝐴 ∈ (𝐽 fLim 𝑔)) → 𝐴 ∈ (𝐽 fClus 𝐹)))
87	73, 86	impbid 212	1 ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝐴 ∈ (𝐽 fClus 𝐹) ↔ ∃𝑔 ∈ (Fil‘𝑋)(𝐹 ⊆ 𝑔 ∧ 𝐴 ∈ (𝐽 fLim 𝑔))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1539   ∈ wcel 2107   ≠ wne 2939  ∀wral 3060  ∃wrex 3069  Vcvv 3479   ∪ cun 3948   ∩ cin 3949   ⊆ wss 3950  ∅c0 4332  𝒫 cpw 4599  {csn 4625  ∪ cuni 4906  ‘cfv 6560  (class class class)co 7432  ficfi 9451  fBascfbas 21353  filGencfg 21354  Topctop 22900  TopOnctopon 22917  neicnei 23106  Filcfil 23854   fLim cflim 23943   fClus cfcls 23945

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-iin 4993  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1o 8507  df-2o 8508  df-en 8987  df-fin 8990  df-fi 9452  df-fbas 21362  df-fg 21363  df-top 22901  df-topon 22918  df-cld 23028  df-ntr 23029  df-cls 23030  df-nei 23107  df-fil 23855  df-flim 23948  df-fcls 23950

This theorem is referenced by:  uffclsflim  24040  cnpfcfi  24049