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Theorem fclsfnflim 23531
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 23355 . . . . . . . 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 23515 . . . . . . . . . 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 22611 . . . . . . . . 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 23385 . . . . . . . . . 10 (𝐹 ∈ (Filβ€˜π‘‹) β†’ βˆͺ 𝐹 = 𝑋)
95fclsfil 23514 . . . . . . . . . . 11 (𝐴 ∈ (𝐽 fClus 𝐹) β†’ 𝐹 ∈ (Filβ€˜βˆͺ 𝐽))
10 filunibas 23385 . . . . . . . . . . 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 4620 . . . . . . . 8 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) β†’ 𝒫 𝑋 = 𝒫 βˆͺ 𝐽)
147, 13sseqtrrd 4024 . . . . . . 7 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) β†’ ((neiβ€˜π½)β€˜{𝐴}) βŠ† 𝒫 𝑋)
152, 14unssd 4187 . . . . . 6 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) β†’ (𝐹 βˆͺ ((neiβ€˜π½)β€˜{𝐴})) βŠ† 𝒫 𝑋)
16 ssun1 4173 . . . . . . . 8 𝐹 βŠ† (𝐹 βˆͺ ((neiβ€˜π½)β€˜{𝐴}))
17 filn0 23366 . . . . . . . 8 (𝐹 ∈ (Filβ€˜π‘‹) β†’ 𝐹 β‰  βˆ…)
18 ssn0 4401 . . . . . . . 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 4202 . . . . . . . . . . . 12 (𝑦 ∩ π‘₯) = (π‘₯ ∩ 𝑦)
22 fclsneii 23521 . . . . . . . . . . . 12 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑦 ∈ ((neiβ€˜π½)β€˜{𝐴}) ∧ π‘₯ ∈ 𝐹) β†’ (𝑦 ∩ π‘₯) β‰  βˆ…)
2321, 22eqnetrrid 3017 . . . . . . . . . . 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 3201 . . . . . . 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 23352 . . . . . . . . 9 (𝐹 ∈ (Filβ€˜π‘‹) β†’ 𝐹 ∈ (fBasβ€˜π‘‹))
2928adantr 482 . . . . . . . 8 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) β†’ 𝐹 ∈ (fBasβ€˜π‘‹))
30 istopon 22414 . . . . . . . . . . 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 23520 . . . . . . . . . . . . 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 4813 . . . . . . . . . 10 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) β†’ {𝐴} βŠ† 𝑋)
36 snnzg 4779 . . . . . . . . . . 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 23384 . . . . . . . . . 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 23352 . . . . . . . . 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 23373 . . . . . . . 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 23359 . . . . . . . 8 (𝐹 ∈ (Filβ€˜π‘‹) β†’ 𝑋 ∈ 𝐹)
46 fsubbas 23371 . . . . . . . 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 23382 . . . . 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 6905 . . . . . . . . 9 ((neiβ€˜π½)β€˜{𝐴}) ∈ V
53 unexg 7736 . . . . . . . . 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 9414 . . . . . . . 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 4188 . . . . 5 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) β†’ 𝐹 βŠ† (fiβ€˜(𝐹 βˆͺ ((neiβ€˜π½)β€˜{𝐴}))))
59 ssfg 23376 . . . . . 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 3993 . . . 4 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) β†’ 𝐹 βŠ† (𝑋filGen(fiβ€˜(𝐹 βˆͺ ((neiβ€˜π½)β€˜{𝐴})))))
6257unssbd 4189 . . . . . 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 3993 . . . . 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 23475 . . . . . 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 4009 . . . . . 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 7417 . . . . . . 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 3613 . . . 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 23474 . . . . . . 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 23527 . . . . 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 23530 . . . . 5 (𝐽 fLim 𝑔) βŠ† (𝐽 fClus 𝑔)
8483, 75sselid 3981 . . . 4 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝑔 ∈ (Filβ€˜π‘‹) ∧ (𝐹 βŠ† 𝑔 ∧ 𝐴 ∈ (𝐽 fLim 𝑔)))) β†’ 𝐴 ∈ (𝐽 fClus 𝑔))
8582, 84sseldd 3984 . . 3 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝑔 ∈ (Filβ€˜π‘‹) ∧ (𝐹 βŠ† 𝑔 ∧ 𝐴 ∈ (𝐽 fLim 𝑔)))) β†’ 𝐴 ∈ (𝐽 fClus 𝐹))
8685rexlimdvaa 3157 . 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 2941  βˆ€wral 3062  βˆƒwrex 3071  Vcvv 3475   βˆͺ cun 3947   ∩ cin 3948   βŠ† wss 3949  βˆ…c0 4323  π’« cpw 4603  {csn 4629  βˆͺ cuni 4909  β€˜cfv 6544  (class class class)co 7409  ficfi 9405  fBascfbas 20932  filGencfg 20933  Topctop 22395  TopOnctopon 22412  neicnei 22601  Filcfil 23349   fLim cflim 23438   fClus cfcls 23440
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 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
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 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-iin 5001  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1o 8466  df-er 8703  df-en 8940  df-fin 8943  df-fi 9406  df-fbas 20941  df-fg 20942  df-top 22396  df-topon 22413  df-cld 22523  df-ntr 22524  df-cls 22525  df-nei 22602  df-fil 23350  df-flim 23443  df-fcls 23445
This theorem is referenced by:  uffclsflim  23535  cnpfcfi  23544
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