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Theorem fclsfnflim 23394
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 ∈ (Filβ€˜π‘‹) β†’ (𝐴 ∈ (𝐽 fClus 𝐹) ↔ βˆƒπ‘” ∈ (Filβ€˜π‘‹)(𝐹 βŠ† 𝑔 ∧ 𝐴 ∈ (𝐽 fLim 𝑔))))
Distinct variable groups:   𝐴,𝑔   𝑔,𝐹   𝑔,𝐽   𝑔,𝑋

Proof of Theorem fclsfnflim
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 filsspw 23218 . . . . . . . 8 (𝐹 ∈ (Filβ€˜π‘‹) β†’ 𝐹 βŠ† 𝒫 𝑋)
21adantr 482 . . . . . . 7 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) β†’ 𝐹 βŠ† 𝒫 𝑋)
3 fclstop 23378 . . . . . . . . . 10 (𝐴 ∈ (𝐽 fClus 𝐹) β†’ 𝐽 ∈ Top)
43adantl 483 . . . . . . . . 9 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) β†’ 𝐽 ∈ Top)
5 eqid 2733 . . . . . . . . . 10 βˆͺ 𝐽 = βˆͺ 𝐽
65neisspw 22474 . . . . . . . . 9 (𝐽 ∈ Top β†’ ((neiβ€˜π½)β€˜{𝐴}) βŠ† 𝒫 βˆͺ 𝐽)
74, 6syl 17 . . . . . . . 8 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) β†’ ((neiβ€˜π½)β€˜{𝐴}) βŠ† 𝒫 βˆͺ 𝐽)
8 filunibas 23248 . . . . . . . . . 10 (𝐹 ∈ (Filβ€˜π‘‹) β†’ βˆͺ 𝐹 = 𝑋)
95fclsfil 23377 . . . . . . . . . . 11 (𝐴 ∈ (𝐽 fClus 𝐹) β†’ 𝐹 ∈ (Filβ€˜βˆͺ 𝐽))
10 filunibas 23248 . . . . . . . . . . 11 (𝐹 ∈ (Filβ€˜βˆͺ 𝐽) β†’ βˆͺ 𝐹 = βˆͺ 𝐽)
119, 10syl 17 . . . . . . . . . 10 (𝐴 ∈ (𝐽 fClus 𝐹) β†’ βˆͺ 𝐹 = βˆͺ 𝐽)
128, 11sylan9req 2794 . . . . . . . . 9 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) β†’ 𝑋 = βˆͺ 𝐽)
1312pweqd 4578 . . . . . . . 8 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) β†’ 𝒫 𝑋 = 𝒫 βˆͺ 𝐽)
147, 13sseqtrrd 3986 . . . . . . 7 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) β†’ ((neiβ€˜π½)β€˜{𝐴}) βŠ† 𝒫 𝑋)
152, 14unssd 4147 . . . . . 6 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) β†’ (𝐹 βˆͺ ((neiβ€˜π½)β€˜{𝐴})) βŠ† 𝒫 𝑋)
16 ssun1 4133 . . . . . . . 8 𝐹 βŠ† (𝐹 βˆͺ ((neiβ€˜π½)β€˜{𝐴}))
17 filn0 23229 . . . . . . . 8 (𝐹 ∈ (Filβ€˜π‘‹) β†’ 𝐹 β‰  βˆ…)
18 ssn0 4361 . . . . . . . 8 ((𝐹 βŠ† (𝐹 βˆͺ ((neiβ€˜π½)β€˜{𝐴})) ∧ 𝐹 β‰  βˆ…) β†’ (𝐹 βˆͺ ((neiβ€˜π½)β€˜{𝐴})) β‰  βˆ…)
1916, 17, 18sylancr 588 . . . . . . 7 (𝐹 ∈ (Filβ€˜π‘‹) β†’ (𝐹 βˆͺ ((neiβ€˜π½)β€˜{𝐴})) β‰  βˆ…)
2019adantr 482 . . . . . 6 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) β†’ (𝐹 βˆͺ ((neiβ€˜π½)β€˜{𝐴})) β‰  βˆ…)
21 incom 4162 . . . . . . . . . . . 12 (𝑦 ∩ π‘₯) = (π‘₯ ∩ 𝑦)
22 fclsneii 23384 . . . . . . . . . . . 12 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑦 ∈ ((neiβ€˜π½)β€˜{𝐴}) ∧ π‘₯ ∈ 𝐹) β†’ (𝑦 ∩ π‘₯) β‰  βˆ…)
2321, 22eqnetrrid 3016 . . . . . . . . . . 11 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑦 ∈ ((neiβ€˜π½)β€˜{𝐴}) ∧ π‘₯ ∈ 𝐹) β†’ (π‘₯ ∩ 𝑦) β‰  βˆ…)
24233com23 1127 . . . . . . . . . 10 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ π‘₯ ∈ 𝐹 ∧ 𝑦 ∈ ((neiβ€˜π½)β€˜{𝐴})) β†’ (π‘₯ ∩ 𝑦) β‰  βˆ…)
25243expb 1121 . . . . . . . . 9 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ (π‘₯ ∈ 𝐹 ∧ 𝑦 ∈ ((neiβ€˜π½)β€˜{𝐴}))) β†’ (π‘₯ ∩ 𝑦) β‰  βˆ…)
2625adantll 713 . . . . . . . 8 (((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) ∧ (π‘₯ ∈ 𝐹 ∧ 𝑦 ∈ ((neiβ€˜π½)β€˜{𝐴}))) β†’ (π‘₯ ∩ 𝑦) β‰  βˆ…)
2726ralrimivva 3194 . . . . . . 7 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) β†’ βˆ€π‘₯ ∈ 𝐹 βˆ€π‘¦ ∈ ((neiβ€˜π½)β€˜{𝐴})(π‘₯ ∩ 𝑦) β‰  βˆ…)
28 filfbas 23215 . . . . . . . . 9 (𝐹 ∈ (Filβ€˜π‘‹) β†’ 𝐹 ∈ (fBasβ€˜π‘‹))
2928adantr 482 . . . . . . . 8 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) β†’ 𝐹 ∈ (fBasβ€˜π‘‹))
30 istopon 22277 . . . . . . . . . . 11 (𝐽 ∈ (TopOnβ€˜π‘‹) ↔ (𝐽 ∈ Top ∧ 𝑋 = βˆͺ 𝐽))
314, 12, 30sylanbrc 584 . . . . . . . . . 10 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
325fclselbas 23383 . . . . . . . . . . . . 13 (𝐴 ∈ (𝐽 fClus 𝐹) β†’ 𝐴 ∈ βˆͺ 𝐽)
3332adantl 483 . . . . . . . . . . . 12 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) β†’ 𝐴 ∈ βˆͺ 𝐽)
3433, 12eleqtrrd 2837 . . . . . . . . . . 11 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) β†’ 𝐴 ∈ 𝑋)
3534snssd 4770 . . . . . . . . . 10 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) β†’ {𝐴} βŠ† 𝑋)
36 snnzg 4736 . . . . . . . . . . 11 (𝐴 ∈ (𝐽 fClus 𝐹) β†’ {𝐴} β‰  βˆ…)
3736adantl 483 . . . . . . . . . 10 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) β†’ {𝐴} β‰  βˆ…)
38 neifil 23247 . . . . . . . . . 10 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ {𝐴} βŠ† 𝑋 ∧ {𝐴} β‰  βˆ…) β†’ ((neiβ€˜π½)β€˜{𝐴}) ∈ (Filβ€˜π‘‹))
3931, 35, 37, 38syl3anc 1372 . . . . . . . . 9 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) β†’ ((neiβ€˜π½)β€˜{𝐴}) ∈ (Filβ€˜π‘‹))
40 filfbas 23215 . . . . . . . . 9 (((neiβ€˜π½)β€˜{𝐴}) ∈ (Filβ€˜π‘‹) β†’ ((neiβ€˜π½)β€˜{𝐴}) ∈ (fBasβ€˜π‘‹))
4139, 40syl 17 . . . . . . . 8 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) β†’ ((neiβ€˜π½)β€˜{𝐴}) ∈ (fBasβ€˜π‘‹))
42 fbunfip 23236 . . . . . . . 8 ((𝐹 ∈ (fBasβ€˜π‘‹) ∧ ((neiβ€˜π½)β€˜{𝐴}) ∈ (fBasβ€˜π‘‹)) β†’ (Β¬ βˆ… ∈ (fiβ€˜(𝐹 βˆͺ ((neiβ€˜π½)β€˜{𝐴}))) ↔ βˆ€π‘₯ ∈ 𝐹 βˆ€π‘¦ ∈ ((neiβ€˜π½)β€˜{𝐴})(π‘₯ ∩ 𝑦) β‰  βˆ…))
4329, 41, 42syl2anc 585 . . . . . . 7 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) β†’ (Β¬ βˆ… ∈ (fiβ€˜(𝐹 βˆͺ ((neiβ€˜π½)β€˜{𝐴}))) ↔ βˆ€π‘₯ ∈ 𝐹 βˆ€π‘¦ ∈ ((neiβ€˜π½)β€˜{𝐴})(π‘₯ ∩ 𝑦) β‰  βˆ…))
4427, 43mpbird 257 . . . . . 6 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) β†’ Β¬ βˆ… ∈ (fiβ€˜(𝐹 βˆͺ ((neiβ€˜π½)β€˜{𝐴}))))
45 filtop 23222 . . . . . . . 8 (𝐹 ∈ (Filβ€˜π‘‹) β†’ 𝑋 ∈ 𝐹)
46 fsubbas 23234 . . . . . . . 8 (𝑋 ∈ 𝐹 β†’ ((fiβ€˜(𝐹 βˆͺ ((neiβ€˜π½)β€˜{𝐴}))) ∈ (fBasβ€˜π‘‹) ↔ ((𝐹 βˆͺ ((neiβ€˜π½)β€˜{𝐴})) βŠ† 𝒫 𝑋 ∧ (𝐹 βˆͺ ((neiβ€˜π½)β€˜{𝐴})) β‰  βˆ… ∧ Β¬ βˆ… ∈ (fiβ€˜(𝐹 βˆͺ ((neiβ€˜π½)β€˜{𝐴}))))))
4745, 46syl 17 . . . . . . 7 (𝐹 ∈ (Filβ€˜π‘‹) β†’ ((fiβ€˜(𝐹 βˆͺ ((neiβ€˜π½)β€˜{𝐴}))) ∈ (fBasβ€˜π‘‹) ↔ ((𝐹 βˆͺ ((neiβ€˜π½)β€˜{𝐴})) βŠ† 𝒫 𝑋 ∧ (𝐹 βˆͺ ((neiβ€˜π½)β€˜{𝐴})) β‰  βˆ… ∧ Β¬ βˆ… ∈ (fiβ€˜(𝐹 βˆͺ ((neiβ€˜π½)β€˜{𝐴}))))))
4847adantr 482 . . . . . 6 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) β†’ ((fiβ€˜(𝐹 βˆͺ ((neiβ€˜π½)β€˜{𝐴}))) ∈ (fBasβ€˜π‘‹) ↔ ((𝐹 βˆͺ ((neiβ€˜π½)β€˜{𝐴})) βŠ† 𝒫 𝑋 ∧ (𝐹 βˆͺ ((neiβ€˜π½)β€˜{𝐴})) β‰  βˆ… ∧ Β¬ βˆ… ∈ (fiβ€˜(𝐹 βˆͺ ((neiβ€˜π½)β€˜{𝐴}))))))
4915, 20, 44, 48mpbir3and 1343 . . . . 5 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) β†’ (fiβ€˜(𝐹 βˆͺ ((neiβ€˜π½)β€˜{𝐴}))) ∈ (fBasβ€˜π‘‹))
50 fgcl 23245 . . . . 5 ((fiβ€˜(𝐹 βˆͺ ((neiβ€˜π½)β€˜{𝐴}))) ∈ (fBasβ€˜π‘‹) β†’ (𝑋filGen(fiβ€˜(𝐹 βˆͺ ((neiβ€˜π½)β€˜{𝐴})))) ∈ (Filβ€˜π‘‹))
5149, 50syl 17 . . . 4 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) β†’ (𝑋filGen(fiβ€˜(𝐹 βˆͺ ((neiβ€˜π½)β€˜{𝐴})))) ∈ (Filβ€˜π‘‹))
52 fvex 6856 . . . . . . . . 9 ((neiβ€˜π½)β€˜{𝐴}) ∈ V
53 unexg 7684 . . . . . . . . 9 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ ((neiβ€˜π½)β€˜{𝐴}) ∈ V) β†’ (𝐹 βˆͺ ((neiβ€˜π½)β€˜{𝐴})) ∈ V)
5452, 53mpan2 690 . . . . . . . 8 (𝐹 ∈ (Filβ€˜π‘‹) β†’ (𝐹 βˆͺ ((neiβ€˜π½)β€˜{𝐴})) ∈ V)
55 ssfii 9360 . . . . . . . 8 ((𝐹 βˆͺ ((neiβ€˜π½)β€˜{𝐴})) ∈ V β†’ (𝐹 βˆͺ ((neiβ€˜π½)β€˜{𝐴})) βŠ† (fiβ€˜(𝐹 βˆͺ ((neiβ€˜π½)β€˜{𝐴}))))
5654, 55syl 17 . . . . . . 7 (𝐹 ∈ (Filβ€˜π‘‹) β†’ (𝐹 βˆͺ ((neiβ€˜π½)β€˜{𝐴})) βŠ† (fiβ€˜(𝐹 βˆͺ ((neiβ€˜π½)β€˜{𝐴}))))
5756adantr 482 . . . . . 6 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) β†’ (𝐹 βˆͺ ((neiβ€˜π½)β€˜{𝐴})) βŠ† (fiβ€˜(𝐹 βˆͺ ((neiβ€˜π½)β€˜{𝐴}))))
5857unssad 4148 . . . . 5 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) β†’ 𝐹 βŠ† (fiβ€˜(𝐹 βˆͺ ((neiβ€˜π½)β€˜{𝐴}))))
59 ssfg 23239 . . . . . 6 ((fiβ€˜(𝐹 βˆͺ ((neiβ€˜π½)β€˜{𝐴}))) ∈ (fBasβ€˜π‘‹) β†’ (fiβ€˜(𝐹 βˆͺ ((neiβ€˜π½)β€˜{𝐴}))) βŠ† (𝑋filGen(fiβ€˜(𝐹 βˆͺ ((neiβ€˜π½)β€˜{𝐴})))))
6049, 59syl 17 . . . . 5 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) β†’ (fiβ€˜(𝐹 βˆͺ ((neiβ€˜π½)β€˜{𝐴}))) βŠ† (𝑋filGen(fiβ€˜(𝐹 βˆͺ ((neiβ€˜π½)β€˜{𝐴})))))
6158, 60sstrd 3955 . . . 4 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) β†’ 𝐹 βŠ† (𝑋filGen(fiβ€˜(𝐹 βˆͺ ((neiβ€˜π½)β€˜{𝐴})))))
6257unssbd 4149 . . . . . 6 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) β†’ ((neiβ€˜π½)β€˜{𝐴}) βŠ† (fiβ€˜(𝐹 βˆͺ ((neiβ€˜π½)β€˜{𝐴}))))
6362, 60sstrd 3955 . . . . 5 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) β†’ ((neiβ€˜π½)β€˜{𝐴}) βŠ† (𝑋filGen(fiβ€˜(𝐹 βˆͺ ((neiβ€˜π½)β€˜{𝐴})))))
64 elflim 23338 . . . . . 6 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (𝑋filGen(fiβ€˜(𝐹 βˆͺ ((neiβ€˜π½)β€˜{𝐴})))) ∈ (Filβ€˜π‘‹)) β†’ (𝐴 ∈ (𝐽 fLim (𝑋filGen(fiβ€˜(𝐹 βˆͺ ((neiβ€˜π½)β€˜{𝐴}))))) ↔ (𝐴 ∈ 𝑋 ∧ ((neiβ€˜π½)β€˜{𝐴}) βŠ† (𝑋filGen(fiβ€˜(𝐹 βˆͺ ((neiβ€˜π½)β€˜{𝐴})))))))
6531, 51, 64syl2anc 585 . . . . 5 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) β†’ (𝐴 ∈ (𝐽 fLim (𝑋filGen(fiβ€˜(𝐹 βˆͺ ((neiβ€˜π½)β€˜{𝐴}))))) ↔ (𝐴 ∈ 𝑋 ∧ ((neiβ€˜π½)β€˜{𝐴}) βŠ† (𝑋filGen(fiβ€˜(𝐹 βˆͺ ((neiβ€˜π½)β€˜{𝐴})))))))
6634, 63, 65mpbir2and 712 . . . 4 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) β†’ 𝐴 ∈ (𝐽 fLim (𝑋filGen(fiβ€˜(𝐹 βˆͺ ((neiβ€˜π½)β€˜{𝐴}))))))
67 sseq2 3971 . . . . . 6 (𝑔 = (𝑋filGen(fiβ€˜(𝐹 βˆͺ ((neiβ€˜π½)β€˜{𝐴})))) β†’ (𝐹 βŠ† 𝑔 ↔ 𝐹 βŠ† (𝑋filGen(fiβ€˜(𝐹 βˆͺ ((neiβ€˜π½)β€˜{𝐴}))))))
68 oveq2 7366 . . . . . . 7 (𝑔 = (𝑋filGen(fiβ€˜(𝐹 βˆͺ ((neiβ€˜π½)β€˜{𝐴})))) β†’ (𝐽 fLim 𝑔) = (𝐽 fLim (𝑋filGen(fiβ€˜(𝐹 βˆͺ ((neiβ€˜π½)β€˜{𝐴}))))))
6968eleq2d 2820 . . . . . 6 (𝑔 = (𝑋filGen(fiβ€˜(𝐹 βˆͺ ((neiβ€˜π½)β€˜{𝐴})))) β†’ (𝐴 ∈ (𝐽 fLim 𝑔) ↔ 𝐴 ∈ (𝐽 fLim (𝑋filGen(fiβ€˜(𝐹 βˆͺ ((neiβ€˜π½)β€˜{𝐴})))))))
7067, 69anbi12d 632 . . . . 5 (𝑔 = (𝑋filGen(fiβ€˜(𝐹 βˆͺ ((neiβ€˜π½)β€˜{𝐴})))) β†’ ((𝐹 βŠ† 𝑔 ∧ 𝐴 ∈ (𝐽 fLim 𝑔)) ↔ (𝐹 βŠ† (𝑋filGen(fiβ€˜(𝐹 βˆͺ ((neiβ€˜π½)β€˜{𝐴})))) ∧ 𝐴 ∈ (𝐽 fLim (𝑋filGen(fiβ€˜(𝐹 βˆͺ ((neiβ€˜π½)β€˜{𝐴}))))))))
7170rspcev 3580 . . . 4 (((𝑋filGen(fiβ€˜(𝐹 βˆͺ ((neiβ€˜π½)β€˜{𝐴})))) ∈ (Filβ€˜π‘‹) ∧ (𝐹 βŠ† (𝑋filGen(fiβ€˜(𝐹 βˆͺ ((neiβ€˜π½)β€˜{𝐴})))) ∧ 𝐴 ∈ (𝐽 fLim (𝑋filGen(fiβ€˜(𝐹 βˆͺ ((neiβ€˜π½)β€˜{𝐴}))))))) β†’ βˆƒπ‘” ∈ (Filβ€˜π‘‹)(𝐹 βŠ† 𝑔 ∧ 𝐴 ∈ (𝐽 fLim 𝑔)))
7251, 61, 66, 71syl12anc 836 . . 3 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) β†’ βˆƒπ‘” ∈ (Filβ€˜π‘‹)(𝐹 βŠ† 𝑔 ∧ 𝐴 ∈ (𝐽 fLim 𝑔)))
7372ex 414 . 2 (𝐹 ∈ (Filβ€˜π‘‹) β†’ (𝐴 ∈ (𝐽 fClus 𝐹) β†’ βˆƒπ‘” ∈ (Filβ€˜π‘‹)(𝐹 βŠ† 𝑔 ∧ 𝐴 ∈ (𝐽 fLim 𝑔))))
74 simprl 770 . . . . . 6 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝑔 ∈ (Filβ€˜π‘‹) ∧ (𝐹 βŠ† 𝑔 ∧ 𝐴 ∈ (𝐽 fLim 𝑔)))) β†’ 𝑔 ∈ (Filβ€˜π‘‹))
75 simprrr 781 . . . . . . 7 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝑔 ∈ (Filβ€˜π‘‹) ∧ (𝐹 βŠ† 𝑔 ∧ 𝐴 ∈ (𝐽 fLim 𝑔)))) β†’ 𝐴 ∈ (𝐽 fLim 𝑔))
76 flimtopon 23337 . . . . . . 7 (𝐴 ∈ (𝐽 fLim 𝑔) β†’ (𝐽 ∈ (TopOnβ€˜π‘‹) ↔ 𝑔 ∈ (Filβ€˜π‘‹)))
7775, 76syl 17 . . . . . 6 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝑔 ∈ (Filβ€˜π‘‹) ∧ (𝐹 βŠ† 𝑔 ∧ 𝐴 ∈ (𝐽 fLim 𝑔)))) β†’ (𝐽 ∈ (TopOnβ€˜π‘‹) ↔ 𝑔 ∈ (Filβ€˜π‘‹)))
7874, 77mpbird 257 . . . . 5 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝑔 ∈ (Filβ€˜π‘‹) ∧ (𝐹 βŠ† 𝑔 ∧ 𝐴 ∈ (𝐽 fLim 𝑔)))) β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
79 simpl 484 . . . . 5 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝑔 ∈ (Filβ€˜π‘‹) ∧ (𝐹 βŠ† 𝑔 ∧ 𝐴 ∈ (𝐽 fLim 𝑔)))) β†’ 𝐹 ∈ (Filβ€˜π‘‹))
80 simprrl 780 . . . . 5 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝑔 ∈ (Filβ€˜π‘‹) ∧ (𝐹 βŠ† 𝑔 ∧ 𝐴 ∈ (𝐽 fLim 𝑔)))) β†’ 𝐹 βŠ† 𝑔)
81 fclsss2 23390 . . . . 5 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐹 βŠ† 𝑔) β†’ (𝐽 fClus 𝑔) βŠ† (𝐽 fClus 𝐹))
8278, 79, 80, 81syl3anc 1372 . . . 4 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝑔 ∈ (Filβ€˜π‘‹) ∧ (𝐹 βŠ† 𝑔 ∧ 𝐴 ∈ (𝐽 fLim 𝑔)))) β†’ (𝐽 fClus 𝑔) βŠ† (𝐽 fClus 𝐹))
83 flimfcls 23393 . . . . 5 (𝐽 fLim 𝑔) βŠ† (𝐽 fClus 𝑔)
8483, 75sselid 3943 . . . 4 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝑔 ∈ (Filβ€˜π‘‹) ∧ (𝐹 βŠ† 𝑔 ∧ 𝐴 ∈ (𝐽 fLim 𝑔)))) β†’ 𝐴 ∈ (𝐽 fClus 𝑔))
8582, 84sseldd 3946 . . 3 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝑔 ∈ (Filβ€˜π‘‹) ∧ (𝐹 βŠ† 𝑔 ∧ 𝐴 ∈ (𝐽 fLim 𝑔)))) β†’ 𝐴 ∈ (𝐽 fClus 𝐹))
8685rexlimdvaa 3150 . 2 (𝐹 ∈ (Filβ€˜π‘‹) β†’ (βˆƒπ‘” ∈ (Filβ€˜π‘‹)(𝐹 βŠ† 𝑔 ∧ 𝐴 ∈ (𝐽 fLim 𝑔)) β†’ 𝐴 ∈ (𝐽 fClus 𝐹)))
8773, 86impbid 211 1 (𝐹 ∈ (Filβ€˜π‘‹) β†’ (𝐴 ∈ (𝐽 fClus 𝐹) ↔ βˆƒπ‘” ∈ (Filβ€˜π‘‹)(𝐹 βŠ† 𝑔 ∧ 𝐴 ∈ (𝐽 fLim 𝑔))))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2940  βˆ€wral 3061  βˆƒwrex 3070  Vcvv 3444   βˆͺ cun 3909   ∩ cin 3910   βŠ† wss 3911  βˆ…c0 4283  π’« cpw 4561  {csn 4587  βˆͺ cuni 4866  β€˜cfv 6497  (class class class)co 7358  ficfi 9351  fBascfbas 20800  filGencfg 20801  Topctop 22258  TopOnctopon 22275  neicnei 22464  Filcfil 23212   fLim cflim 23301   fClus cfcls 23303
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-iin 4958  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-1o 8413  df-er 8651  df-en 8887  df-fin 8890  df-fi 9352  df-fbas 20809  df-fg 20810  df-top 22259  df-topon 22276  df-cld 22386  df-ntr 22387  df-cls 22388  df-nei 22465  df-fil 23213  df-flim 23306  df-fcls 23308
This theorem is referenced by:  uffclsflim  23398  cnpfcfi  23407
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