Theorem fclsfnflim 23912

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 23736	. . . . . . . 8 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ⊆ 𝒫 𝑋)
2	1	adantr 480	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → 𝐹 ⊆ 𝒫 𝑋)
3		fclstop 23896	. . . . . . . . . 10 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐽 ∈ Top)
4	3	adantl 481	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → 𝐽 ∈ Top)
5		eqid 2729	. . . . . . . . . 10 ⊢ ∪ 𝐽 = ∪ 𝐽
6	5	neisspw 22992	. . . . . . . . 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 23766	. . . . . . . . . 10 ⊢ (𝐹 ∈ (Fil‘𝑋) → ∪ 𝐹 = 𝑋)
9	5	fclsfil 23895	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐹 ∈ (Fil‘∪ 𝐽))
10		filunibas 23766	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (Fil‘∪ 𝐽) → ∪ 𝐹 = ∪ 𝐽)
11	9, 10	syl 17	. . . . . . . . . 10 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → ∪ 𝐹 = ∪ 𝐽)
12	8, 11	sylan9req 2785	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → 𝑋 = ∪ 𝐽)
13	12	pweqd 4568	. . . . . . . 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 4143	. . . . . 6 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → (𝐹 ∪ ((nei‘𝐽)‘{𝐴})) ⊆ 𝒫 𝑋)
16		ssun1 4129	. . . . . . . 8 ⊢ 𝐹 ⊆ (𝐹 ∪ ((nei‘𝐽)‘{𝐴}))
17		filn0 23747	. . . . . . . 8 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ≠ ∅)
18		ssn0 4355	. . . . . . . 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 4160	. . . . . . . . . . . 12 ⊢ (𝑦 ∩ 𝑥) = (𝑥 ∩ 𝑦)
22		fclsneii 23902	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑥 ∈ 𝐹) → (𝑦 ∩ 𝑥) ≠ ∅)
23	21, 22	eqnetrrid 3000	. . . . . . . . . . 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 3172	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ ((nei‘𝐽)‘{𝐴})(𝑥 ∩ 𝑦) ≠ ∅)
28		filfbas 23733	. . . . . . . . 9 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋))
29	28	adantr 480	. . . . . . . 8 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → 𝐹 ∈ (fBas‘𝑋))
30		istopon 22797	. . . . . . . . . . 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 23901	. . . . . . . . . . . . 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 2831	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → 𝐴 ∈ 𝑋)
35	34	snssd 4760	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → {𝐴} ⊆ 𝑋)
36		snnzg 4726	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → {𝐴} ≠ ∅)
37	36	adantl 481	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → {𝐴} ≠ ∅)
38		neifil 23765	. . . . . . . . . 10 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ {𝐴} ⊆ 𝑋 ∧ {𝐴} ≠ ∅) → ((nei‘𝐽)‘{𝐴}) ∈ (Fil‘𝑋))
39	31, 35, 37, 38	syl3anc 1373	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → ((nei‘𝐽)‘{𝐴}) ∈ (Fil‘𝑋))
40		filfbas 23733	. . . . . . . . 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 23754	. . . . . . . 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 23740	. . . . . . . 8 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝑋 ∈ 𝐹)
46		fsubbas 23752	. . . . . . . 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 1343	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → (fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴}))) ∈ (fBas‘𝑋))
50		fgcl 23763	. . . . 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 6835	. . . . . . . . 9 ⊢ ((nei‘𝐽)‘{𝐴}) ∈ V
53		unexg 7679	. . . . . . . . 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 9309	. . . . . . . 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 4144	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → 𝐹 ⊆ (fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴}))))
59		ssfg 23757	. . . . . 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 4145	. . . . . 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 23856	. . . . . 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 3962	. . . . . 6 ⊢ (𝑔 = (𝑋filGen(fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴})))) → (𝐹 ⊆ 𝑔 ↔ 𝐹 ⊆ (𝑋filGen(fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴}))))))
68		oveq2 7357	. . . . . . 7 ⊢ (𝑔 = (𝑋filGen(fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴})))) → (𝐽 fLim 𝑔) = (𝐽 fLim (𝑋filGen(fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴}))))))
69	68	eleq2d 2814	. . . . . 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 3577	. . . 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 23855	. . . . . . 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 23908	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝑔) → (𝐽 fClus 𝑔) ⊆ (𝐽 fClus 𝐹))
82	78, 79, 80, 81	syl3anc 1373	. . . 4 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ (𝐹 ⊆ 𝑔 ∧ 𝐴 ∈ (𝐽 fLim 𝑔)))) → (𝐽 fClus 𝑔) ⊆ (𝐽 fClus 𝐹))
83		flimfcls 23911	. . . . 5 ⊢ (𝐽 fLim 𝑔) ⊆ (𝐽 fClus 𝑔)
84	83, 75	sselid 3933	. . . 4 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ (𝐹 ⊆ 𝑔 ∧ 𝐴 ∈ (𝐽 fLim 𝑔)))) → 𝐴 ∈ (𝐽 fClus 𝑔))
85	82, 84	sseldd 3936	. . 3 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ (𝐹 ⊆ 𝑔 ∧ 𝐴 ∈ (𝐽 fLim 𝑔)))) → 𝐴 ∈ (𝐽 fClus 𝐹))
86	85	rexlimdvaa 3131	. 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 1540   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  ∃wrex 3053  Vcvv 3436   ∪ cun 3901   ∩ cin 3902   ⊆ wss 3903  ∅c0 4284  𝒫 cpw 4551  {csn 4577  ∪ cuni 4858  ‘cfv 6482  (class class class)co 7349  ficfi 9300  fBascfbas 21249  filGencfg 21250  Topctop 22778  TopOnctopon 22795  neicnei 22982  Filcfil 23730   fLim cflim 23819   fClus cfcls 23821

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-int 4897  df-iun 4943  df-iin 4944  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-1o 8388  df-2o 8389  df-en 8873  df-fin 8876  df-fi 9301  df-fbas 21258  df-fg 21259  df-top 22779  df-topon 22796  df-cld 22904  df-ntr 22905  df-cls 22906  df-nei 22983  df-fil 23731  df-flim 23824  df-fcls 23826

This theorem is referenced by:  uffclsflim  23916  cnpfcfi  23925