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Theorem fclsfnflim 21878
 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)(𝐹𝑔𝐴 ∈ (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.
StepHypRef Expression
1 filsspw 21702 . . . . . . . 8 (𝐹 ∈ (Fil‘𝑋) → 𝐹 ⊆ 𝒫 𝑋)
21adantr 480 . . . . . . 7 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → 𝐹 ⊆ 𝒫 𝑋)
3 fclstop 21862 . . . . . . . . . 10 (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐽 ∈ Top)
43adantl 481 . . . . . . . . 9 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → 𝐽 ∈ Top)
5 eqid 2651 . . . . . . . . . 10 𝐽 = 𝐽
65neisspw 20959 . . . . . . . . 9 (𝐽 ∈ Top → ((nei‘𝐽)‘{𝐴}) ⊆ 𝒫 𝐽)
74, 6syl 17 . . . . . . . 8 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → ((nei‘𝐽)‘{𝐴}) ⊆ 𝒫 𝐽)
8 filunibas 21732 . . . . . . . . . 10 (𝐹 ∈ (Fil‘𝑋) → 𝐹 = 𝑋)
95fclsfil 21861 . . . . . . . . . . 11 (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐹 ∈ (Fil‘ 𝐽))
10 filunibas 21732 . . . . . . . . . . 11 (𝐹 ∈ (Fil‘ 𝐽) → 𝐹 = 𝐽)
119, 10syl 17 . . . . . . . . . 10 (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐹 = 𝐽)
128, 11sylan9req 2706 . . . . . . . . 9 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → 𝑋 = 𝐽)
1312pweqd 4196 . . . . . . . 8 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → 𝒫 𝑋 = 𝒫 𝐽)
147, 13sseqtr4d 3675 . . . . . . 7 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → ((nei‘𝐽)‘{𝐴}) ⊆ 𝒫 𝑋)
152, 14unssd 3822 . . . . . 6 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → (𝐹 ∪ ((nei‘𝐽)‘{𝐴})) ⊆ 𝒫 𝑋)
16 ssun1 3809 . . . . . . . 8 𝐹 ⊆ (𝐹 ∪ ((nei‘𝐽)‘{𝐴}))
17 filn0 21713 . . . . . . . 8 (𝐹 ∈ (Fil‘𝑋) → 𝐹 ≠ ∅)
18 ssn0 4009 . . . . . . . 8 ((𝐹 ⊆ (𝐹 ∪ ((nei‘𝐽)‘{𝐴})) ∧ 𝐹 ≠ ∅) → (𝐹 ∪ ((nei‘𝐽)‘{𝐴})) ≠ ∅)
1916, 17, 18sylancr 696 . . . . . . 7 (𝐹 ∈ (Fil‘𝑋) → (𝐹 ∪ ((nei‘𝐽)‘{𝐴})) ≠ ∅)
2019adantr 480 . . . . . 6 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → (𝐹 ∪ ((nei‘𝐽)‘{𝐴})) ≠ ∅)
21 incom 3838 . . . . . . . . . . . 12 (𝑦𝑥) = (𝑥𝑦)
22 fclsneii 21868 . . . . . . . . . . . 12 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑥𝐹) → (𝑦𝑥) ≠ ∅)
2321, 22syl5eqner 2898 . . . . . . . . . . 11 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑥𝐹) → (𝑥𝑦) ≠ ∅)
24233com23 1291 . . . . . . . . . 10 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑥𝐹𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → (𝑥𝑦) ≠ ∅)
25243expb 1285 . . . . . . . . 9 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ (𝑥𝐹𝑦 ∈ ((nei‘𝐽)‘{𝐴}))) → (𝑥𝑦) ≠ ∅)
2625adantll 750 . . . . . . . 8 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) ∧ (𝑥𝐹𝑦 ∈ ((nei‘𝐽)‘{𝐴}))) → (𝑥𝑦) ≠ ∅)
2726ralrimivva 3000 . . . . . . 7 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → ∀𝑥𝐹𝑦 ∈ ((nei‘𝐽)‘{𝐴})(𝑥𝑦) ≠ ∅)
28 filfbas 21699 . . . . . . . . 9 (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋))
2928adantr 480 . . . . . . . 8 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → 𝐹 ∈ (fBas‘𝑋))
30 istopon 20765 . . . . . . . . . . 11 (𝐽 ∈ (TopOn‘𝑋) ↔ (𝐽 ∈ Top ∧ 𝑋 = 𝐽))
314, 12, 30sylanbrc 699 . . . . . . . . . 10 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → 𝐽 ∈ (TopOn‘𝑋))
325fclselbas 21867 . . . . . . . . . . . . 13 (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐴 𝐽)
3332adantl 481 . . . . . . . . . . . 12 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → 𝐴 𝐽)
3433, 12eleqtrrd 2733 . . . . . . . . . . 11 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → 𝐴𝑋)
3534snssd 4372 . . . . . . . . . 10 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → {𝐴} ⊆ 𝑋)
36 snnzg 4339 . . . . . . . . . . 11 (𝐴 ∈ (𝐽 fClus 𝐹) → {𝐴} ≠ ∅)
3736adantl 481 . . . . . . . . . 10 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → {𝐴} ≠ ∅)
38 neifil 21731 . . . . . . . . . 10 ((𝐽 ∈ (TopOn‘𝑋) ∧ {𝐴} ⊆ 𝑋 ∧ {𝐴} ≠ ∅) → ((nei‘𝐽)‘{𝐴}) ∈ (Fil‘𝑋))
3931, 35, 37, 38syl3anc 1366 . . . . . . . . 9 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → ((nei‘𝐽)‘{𝐴}) ∈ (Fil‘𝑋))
40 filfbas 21699 . . . . . . . . 9 (((nei‘𝐽)‘{𝐴}) ∈ (Fil‘𝑋) → ((nei‘𝐽)‘{𝐴}) ∈ (fBas‘𝑋))
4139, 40syl 17 . . . . . . . 8 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → ((nei‘𝐽)‘{𝐴}) ∈ (fBas‘𝑋))
42 fbunfip 21720 . . . . . . . 8 ((𝐹 ∈ (fBas‘𝑋) ∧ ((nei‘𝐽)‘{𝐴}) ∈ (fBas‘𝑋)) → (¬ ∅ ∈ (fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴}))) ↔ ∀𝑥𝐹𝑦 ∈ ((nei‘𝐽)‘{𝐴})(𝑥𝑦) ≠ ∅))
4329, 41, 42syl2anc 694 . . . . . . 7 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → (¬ ∅ ∈ (fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴}))) ↔ ∀𝑥𝐹𝑦 ∈ ((nei‘𝐽)‘{𝐴})(𝑥𝑦) ≠ ∅))
4427, 43mpbird 247 . . . . . 6 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → ¬ ∅ ∈ (fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴}))))
45 filtop 21706 . . . . . . . 8 (𝐹 ∈ (Fil‘𝑋) → 𝑋𝐹)
46 fsubbas 21718 . . . . . . . 8 (𝑋𝐹 → ((fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴}))) ∈ (fBas‘𝑋) ↔ ((𝐹 ∪ ((nei‘𝐽)‘{𝐴})) ⊆ 𝒫 𝑋 ∧ (𝐹 ∪ ((nei‘𝐽)‘{𝐴})) ≠ ∅ ∧ ¬ ∅ ∈ (fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴}))))))
4745, 46syl 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}))))))
4847adantr 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}))))))
4915, 20, 44, 48mpbir3and 1264 . . . . 5 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → (fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴}))) ∈ (fBas‘𝑋))
50 fgcl 21729 . . . . 5 ((fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴}))) ∈ (fBas‘𝑋) → (𝑋filGen(fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴})))) ∈ (Fil‘𝑋))
5149, 50syl 17 . . . 4 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → (𝑋filGen(fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴})))) ∈ (Fil‘𝑋))
52 fvex 6239 . . . . . . . . 9 ((nei‘𝐽)‘{𝐴}) ∈ V
53 unexg 7001 . . . . . . . . 9 ((𝐹 ∈ (Fil‘𝑋) ∧ ((nei‘𝐽)‘{𝐴}) ∈ V) → (𝐹 ∪ ((nei‘𝐽)‘{𝐴})) ∈ V)
5452, 53mpan2 707 . . . . . . . 8 (𝐹 ∈ (Fil‘𝑋) → (𝐹 ∪ ((nei‘𝐽)‘{𝐴})) ∈ V)
55 ssfii 8366 . . . . . . . 8 ((𝐹 ∪ ((nei‘𝐽)‘{𝐴})) ∈ V → (𝐹 ∪ ((nei‘𝐽)‘{𝐴})) ⊆ (fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴}))))
5654, 55syl 17 . . . . . . 7 (𝐹 ∈ (Fil‘𝑋) → (𝐹 ∪ ((nei‘𝐽)‘{𝐴})) ⊆ (fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴}))))
5756adantr 480 . . . . . 6 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → (𝐹 ∪ ((nei‘𝐽)‘{𝐴})) ⊆ (fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴}))))
5857unssad 3823 . . . . 5 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → 𝐹 ⊆ (fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴}))))
59 ssfg 21723 . . . . . 6 ((fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴}))) ∈ (fBas‘𝑋) → (fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴}))) ⊆ (𝑋filGen(fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴})))))
6049, 59syl 17 . . . . 5 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → (fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴}))) ⊆ (𝑋filGen(fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴})))))
6158, 60sstrd 3646 . . . 4 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → 𝐹 ⊆ (𝑋filGen(fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴})))))
6257unssbd 3824 . . . . . 6 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → ((nei‘𝐽)‘{𝐴}) ⊆ (fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴}))))
6362, 60sstrd 3646 . . . . 5 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → ((nei‘𝐽)‘{𝐴}) ⊆ (𝑋filGen(fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴})))))
64 elflim 21822 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑋filGen(fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴})))) ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fLim (𝑋filGen(fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴}))))) ↔ (𝐴𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ (𝑋filGen(fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴})))))))
6531, 51, 64syl2anc 694 . . . . 5 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → (𝐴 ∈ (𝐽 fLim (𝑋filGen(fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴}))))) ↔ (𝐴𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ (𝑋filGen(fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴})))))))
6634, 63, 65mpbir2and 977 . . . 4 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → 𝐴 ∈ (𝐽 fLim (𝑋filGen(fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴}))))))
67 sseq2 3660 . . . . . 6 (𝑔 = (𝑋filGen(fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴})))) → (𝐹𝑔𝐹 ⊆ (𝑋filGen(fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴}))))))
68 oveq2 6698 . . . . . . 7 (𝑔 = (𝑋filGen(fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴})))) → (𝐽 fLim 𝑔) = (𝐽 fLim (𝑋filGen(fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴}))))))
6968eleq2d 2716 . . . . . 6 (𝑔 = (𝑋filGen(fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴})))) → (𝐴 ∈ (𝐽 fLim 𝑔) ↔ 𝐴 ∈ (𝐽 fLim (𝑋filGen(fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴})))))))
7067, 69anbi12d 747 . . . . 5 (𝑔 = (𝑋filGen(fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴})))) → ((𝐹𝑔𝐴 ∈ (𝐽 fLim 𝑔)) ↔ (𝐹 ⊆ (𝑋filGen(fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴})))) ∧ 𝐴 ∈ (𝐽 fLim (𝑋filGen(fi‘(𝐹 ∪ ((nei‘𝐽)‘{𝐴}))))))))
7170rspcev 3340 . . . 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)(𝐹𝑔𝐴 ∈ (𝐽 fLim 𝑔)))
7251, 61, 66, 71syl12anc 1364 . . 3 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) → ∃𝑔 ∈ (Fil‘𝑋)(𝐹𝑔𝐴 ∈ (𝐽 fLim 𝑔)))
7372ex 449 . 2 (𝐹 ∈ (Fil‘𝑋) → (𝐴 ∈ (𝐽 fClus 𝐹) → ∃𝑔 ∈ (Fil‘𝑋)(𝐹𝑔𝐴 ∈ (𝐽 fLim 𝑔))))
74 simprl 809 . . . . . 6 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ (𝐹𝑔𝐴 ∈ (𝐽 fLim 𝑔)))) → 𝑔 ∈ (Fil‘𝑋))
75 simprrr 822 . . . . . . 7 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ (𝐹𝑔𝐴 ∈ (𝐽 fLim 𝑔)))) → 𝐴 ∈ (𝐽 fLim 𝑔))
76 flimtopon 21821 . . . . . . 7 (𝐴 ∈ (𝐽 fLim 𝑔) → (𝐽 ∈ (TopOn‘𝑋) ↔ 𝑔 ∈ (Fil‘𝑋)))
7775, 76syl 17 . . . . . 6 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ (𝐹𝑔𝐴 ∈ (𝐽 fLim 𝑔)))) → (𝐽 ∈ (TopOn‘𝑋) ↔ 𝑔 ∈ (Fil‘𝑋)))
7874, 77mpbird 247 . . . . 5 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ (𝐹𝑔𝐴 ∈ (𝐽 fLim 𝑔)))) → 𝐽 ∈ (TopOn‘𝑋))
79 simpl 472 . . . . 5 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ (𝐹𝑔𝐴 ∈ (𝐽 fLim 𝑔)))) → 𝐹 ∈ (Fil‘𝑋))
80 simprrl 821 . . . . 5 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ (𝐹𝑔𝐴 ∈ (𝐽 fLim 𝑔)))) → 𝐹𝑔)
81 fclsss2 21874 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐹𝑔) → (𝐽 fClus 𝑔) ⊆ (𝐽 fClus 𝐹))
8278, 79, 80, 81syl3anc 1366 . . . 4 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ (𝐹𝑔𝐴 ∈ (𝐽 fLim 𝑔)))) → (𝐽 fClus 𝑔) ⊆ (𝐽 fClus 𝐹))
83 flimfcls 21877 . . . . 5 (𝐽 fLim 𝑔) ⊆ (𝐽 fClus 𝑔)
8483, 75sseldi 3634 . . . 4 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ (𝐹𝑔𝐴 ∈ (𝐽 fLim 𝑔)))) → 𝐴 ∈ (𝐽 fClus 𝑔))
8582, 84sseldd 3637 . . 3 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑔 ∈ (Fil‘𝑋) ∧ (𝐹𝑔𝐴 ∈ (𝐽 fLim 𝑔)))) → 𝐴 ∈ (𝐽 fClus 𝐹))
8685rexlimdvaa 3061 . 2 (𝐹 ∈ (Fil‘𝑋) → (∃𝑔 ∈ (Fil‘𝑋)(𝐹𝑔𝐴 ∈ (𝐽 fLim 𝑔)) → 𝐴 ∈ (𝐽 fClus 𝐹)))
8773, 86impbid 202 1 (𝐹 ∈ (Fil‘𝑋) → (𝐴 ∈ (𝐽 fClus 𝐹) ↔ ∃𝑔 ∈ (Fil‘𝑋)(𝐹𝑔𝐴 ∈ (𝐽 fLim 𝑔))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 383   ∧ w3a 1054   = wceq 1523   ∈ wcel 2030   ≠ wne 2823  ∀wral 2941  ∃wrex 2942  Vcvv 3231   ∪ cun 3605   ∩ cin 3606   ⊆ wss 3607  ∅c0 3948  𝒫 cpw 4191  {csn 4210  ∪ cuni 4468  ‘cfv 5926  (class class class)co 6690  ficfi 8357  fBascfbas 19782  filGencfg 19783  Topctop 20746  TopOnctopon 20763  neicnei 20949  Filcfil 21696   fLim cflim 21785   fClus cfcls 21787 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-en 7998  df-fin 8001  df-fi 8358  df-fbas 19791  df-fg 19792  df-top 20747  df-topon 20764  df-cld 20871  df-ntr 20872  df-cls 20873  df-nei 20950  df-fil 21697  df-flim 21790  df-fcls 21792 This theorem is referenced by:  uffclsflim  21882  cnpfcfi  21891
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