Theorem fclsfnflim 23890

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 23714	. . . . . . . 8 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ⊆ 𝒫 𝑋)
2	1	adantr 480	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → 𝐹 ⊆ 𝒫 𝑋)
3		fclstop 23874	. . . . . . . . . 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 22970	. . . . . . . . 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 23744	. . . . . . . . . 10 ⊢ (𝐹 ∈ (Fil‘𝑋) → ∪ 𝐹 = 𝑋)
9	5	fclsfil 23873	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐹 ∈ (Fil‘∪ 𝐽))
10		filunibas 23744	. . . . . . . . . . 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 4576	. . . . . . . 8 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → 𝒫 𝑋 = 𝒫 ∪ 𝐽)
14	7, 13	sseqtrrd 3981	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → ((nei‘𝐽)‘{𝐴}) ⊆ 𝒫 𝑋)
15	2, 14	unssd 4151	. . . . . 6 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → (𝐹 ∪ ((nei‘𝐽)‘{𝐴})) ⊆ 𝒫 𝑋)
16		ssun1 4137	. . . . . . . 8 ⊢ 𝐹 ⊆ (𝐹 ∪ ((nei‘𝐽)‘{𝐴}))
17		filn0 23725	. . . . . . . 8 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ≠ ∅)
18		ssn0 4363	. . . . . . . 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 4168	. . . . . . . . . . . 12 ⊢ (𝑦 ∩ 𝑥) = (𝑥 ∩ 𝑦)
22		fclsneii 23880	. . . . . . . . . . . 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 3178	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ ((nei‘𝐽)‘{𝐴})(𝑥 ∩ 𝑦) ≠ ∅)
28		filfbas 23711	. . . . . . . . 9 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋))
29	28	adantr 480	. . . . . . . 8 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → 𝐹 ∈ (fBas‘𝑋))
30		istopon 22775	. . . . . . . . . . 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 23879	. . . . . . . . . . . . 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 4769	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → {𝐴} ⊆ 𝑋)
36		snnzg 4734	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → {𝐴} ≠ ∅)
37	36	adantl 481	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → {𝐴} ≠ ∅)
38		neifil 23743	. . . . . . . . . 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 23711	. . . . . . . . 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 23732	. . . . . . . 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 23718	. . . . . . . 8 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝑋 ∈ 𝐹)
46		fsubbas 23730	. . . . . . . 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 23741	. . . . 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 6853	. . . . . . . . 9 ⊢ ((nei‘𝐽)‘{𝐴}) ∈ V
53		unexg 7699	. . . . . . . . 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 9346	. . . . . . . 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 4152	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → 𝐹 ⊆ (fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴}))))
59		ssfg 23735	. . . . . 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 3954	. . . 4 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → 𝐹 ⊆ (𝑋filGen(fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴})))))
62	57	unssbd 4153	. . . . . 6 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → ((nei‘𝐽)‘{𝐴}) ⊆ (fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴}))))
63	62, 60	sstrd 3954	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → ((nei‘𝐽)‘{𝐴}) ⊆ (𝑋filGen(fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴})))))
64		elflim 23834	. . . . . 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 3970	. . . . . 6 ⊢ (𝑔 = (𝑋filGen(fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴})))) → (𝐹 ⊆ 𝑔 ↔ 𝐹 ⊆ (𝑋filGen(fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴}))))))
68		oveq2 7377	. . . . . . 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 3585	. . . 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 23833	. . . . . . 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 23886	. . . . 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 23889	. . . . 5 ⊢ (𝐽 fLim 𝑔) ⊆ (𝐽 fClus 𝑔)
84	83, 75	sselid 3941	. . . 4 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ (𝐹 ⊆ 𝑔 ∧ 𝐴 ∈ (𝐽 fLim 𝑔)))) → 𝐴 ∈ (𝐽 fClus 𝑔))
85	82, 84	sseldd 3944	. . 3 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ (𝐹 ⊆ 𝑔 ∧ 𝐴 ∈ (𝐽 fLim 𝑔)))) → 𝐴 ∈ (𝐽 fClus 𝐹))
86	85	rexlimdvaa 3135	. 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 3444   ∪ cun 3909   ∩ cin 3910   ⊆ wss 3911  ∅c0 4292  𝒫 cpw 4559  {csn 4585  ∪ cuni 4867  ‘cfv 6499  (class class class)co 7369  ficfi 9337  fBascfbas 21228  filGencfg 21229  Topctop 22756  TopOnctopon 22773  neicnei 22960  Filcfil 23708   fLim cflim 23797   fClus cfcls 23799

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 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

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 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-int 4907  df-iun 4953  df-iin 4954  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7823  df-1o 8411  df-2o 8412  df-en 8896  df-fin 8899  df-fi 9338  df-fbas 21237  df-fg 21238  df-top 22757  df-topon 22774  df-cld 22882  df-ntr 22883  df-cls 22884  df-nei 22961  df-fil 23709  df-flim 23802  df-fcls 23804

This theorem is referenced by:  uffclsflim  23894  cnpfcfi  23903