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Theorem fclsfnflim 23522
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 23346 . . . . . . . 8 (𝐹 ∈ (Filβ€˜π‘‹) β†’ 𝐹 βŠ† 𝒫 𝑋)
21adantr 481 . . . . . . 7 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) β†’ 𝐹 βŠ† 𝒫 𝑋)
3 fclstop 23506 . . . . . . . . . 10 (𝐴 ∈ (𝐽 fClus 𝐹) β†’ 𝐽 ∈ Top)
43adantl 482 . . . . . . . . 9 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) β†’ 𝐽 ∈ Top)
5 eqid 2732 . . . . . . . . . 10 βˆͺ 𝐽 = βˆͺ 𝐽
65neisspw 22602 . . . . . . . . 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 23376 . . . . . . . . . 10 (𝐹 ∈ (Filβ€˜π‘‹) β†’ βˆͺ 𝐹 = 𝑋)
95fclsfil 23505 . . . . . . . . . . 11 (𝐴 ∈ (𝐽 fClus 𝐹) β†’ 𝐹 ∈ (Filβ€˜βˆͺ 𝐽))
10 filunibas 23376 . . . . . . . . . . 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 2793 . . . . . . . . 9 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) β†’ 𝑋 = βˆͺ 𝐽)
1312pweqd 4618 . . . . . . . 8 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) β†’ 𝒫 𝑋 = 𝒫 βˆͺ 𝐽)
147, 13sseqtrrd 4022 . . . . . . 7 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) β†’ ((neiβ€˜π½)β€˜{𝐴}) βŠ† 𝒫 𝑋)
152, 14unssd 4185 . . . . . 6 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) β†’ (𝐹 βˆͺ ((neiβ€˜π½)β€˜{𝐴})) βŠ† 𝒫 𝑋)
16 ssun1 4171 . . . . . . . 8 𝐹 βŠ† (𝐹 βˆͺ ((neiβ€˜π½)β€˜{𝐴}))
17 filn0 23357 . . . . . . . 8 (𝐹 ∈ (Filβ€˜π‘‹) β†’ 𝐹 β‰  βˆ…)
18 ssn0 4399 . . . . . . . 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 587 . . . . . . 7 (𝐹 ∈ (Filβ€˜π‘‹) β†’ (𝐹 βˆͺ ((neiβ€˜π½)β€˜{𝐴})) β‰  βˆ…)
2019adantr 481 . . . . . 6 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) β†’ (𝐹 βˆͺ ((neiβ€˜π½)β€˜{𝐴})) β‰  βˆ…)
21 incom 4200 . . . . . . . . . . . 12 (𝑦 ∩ π‘₯) = (π‘₯ ∩ 𝑦)
22 fclsneii 23512 . . . . . . . . . . . 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 1126 . . . . . . . . . 10 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ π‘₯ ∈ 𝐹 ∧ 𝑦 ∈ ((neiβ€˜π½)β€˜{𝐴})) β†’ (π‘₯ ∩ 𝑦) β‰  βˆ…)
25243expb 1120 . . . . . . . . 9 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ (π‘₯ ∈ 𝐹 ∧ 𝑦 ∈ ((neiβ€˜π½)β€˜{𝐴}))) β†’ (π‘₯ ∩ 𝑦) β‰  βˆ…)
2625adantll 712 . . . . . . . 8 (((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) ∧ (π‘₯ ∈ 𝐹 ∧ 𝑦 ∈ ((neiβ€˜π½)β€˜{𝐴}))) β†’ (π‘₯ ∩ 𝑦) β‰  βˆ…)
2726ralrimivva 3200 . . . . . . 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 23343 . . . . . . . . 9 (𝐹 ∈ (Filβ€˜π‘‹) β†’ 𝐹 ∈ (fBasβ€˜π‘‹))
2928adantr 481 . . . . . . . 8 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) β†’ 𝐹 ∈ (fBasβ€˜π‘‹))
30 istopon 22405 . . . . . . . . . . 11 (𝐽 ∈ (TopOnβ€˜π‘‹) ↔ (𝐽 ∈ Top ∧ 𝑋 = βˆͺ 𝐽))
314, 12, 30sylanbrc 583 . . . . . . . . . 10 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
325fclselbas 23511 . . . . . . . . . . . . 13 (𝐴 ∈ (𝐽 fClus 𝐹) β†’ 𝐴 ∈ βˆͺ 𝐽)
3332adantl 482 . . . . . . . . . . . 12 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) β†’ 𝐴 ∈ βˆͺ 𝐽)
3433, 12eleqtrrd 2836 . . . . . . . . . . 11 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) β†’ 𝐴 ∈ 𝑋)
3534snssd 4811 . . . . . . . . . 10 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) β†’ {𝐴} βŠ† 𝑋)
36 snnzg 4777 . . . . . . . . . . 11 (𝐴 ∈ (𝐽 fClus 𝐹) β†’ {𝐴} β‰  βˆ…)
3736adantl 482 . . . . . . . . . 10 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) β†’ {𝐴} β‰  βˆ…)
38 neifil 23375 . . . . . . . . . 10 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ {𝐴} βŠ† 𝑋 ∧ {𝐴} β‰  βˆ…) β†’ ((neiβ€˜π½)β€˜{𝐴}) ∈ (Filβ€˜π‘‹))
3931, 35, 37, 38syl3anc 1371 . . . . . . . . 9 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) β†’ ((neiβ€˜π½)β€˜{𝐴}) ∈ (Filβ€˜π‘‹))
40 filfbas 23343 . . . . . . . . 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 23364 . . . . . . . 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 584 . . . . . . 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 256 . . . . . 6 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) β†’ Β¬ βˆ… ∈ (fiβ€˜(𝐹 βˆͺ ((neiβ€˜π½)β€˜{𝐴}))))
45 filtop 23350 . . . . . . . 8 (𝐹 ∈ (Filβ€˜π‘‹) β†’ 𝑋 ∈ 𝐹)
46 fsubbas 23362 . . . . . . . 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 481 . . . . . 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 1342 . . . . 5 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) β†’ (fiβ€˜(𝐹 βˆͺ ((neiβ€˜π½)β€˜{𝐴}))) ∈ (fBasβ€˜π‘‹))
50 fgcl 23373 . . . . 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 6901 . . . . . . . . 9 ((neiβ€˜π½)β€˜{𝐴}) ∈ V
53 unexg 7732 . . . . . . . . 9 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ ((neiβ€˜π½)β€˜{𝐴}) ∈ V) β†’ (𝐹 βˆͺ ((neiβ€˜π½)β€˜{𝐴})) ∈ V)
5452, 53mpan2 689 . . . . . . . 8 (𝐹 ∈ (Filβ€˜π‘‹) β†’ (𝐹 βˆͺ ((neiβ€˜π½)β€˜{𝐴})) ∈ V)
55 ssfii 9410 . . . . . . . 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 481 . . . . . 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 4186 . . . . 5 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) β†’ 𝐹 βŠ† (fiβ€˜(𝐹 βˆͺ ((neiβ€˜π½)β€˜{𝐴}))))
59 ssfg 23367 . . . . . 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 3991 . . . 4 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) β†’ 𝐹 βŠ† (𝑋filGen(fiβ€˜(𝐹 βˆͺ ((neiβ€˜π½)β€˜{𝐴})))))
6257unssbd 4187 . . . . . 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 3991 . . . . 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 23466 . . . . . 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 584 . . . . 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 711 . . . 4 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) β†’ 𝐴 ∈ (𝐽 fLim (𝑋filGen(fiβ€˜(𝐹 βˆͺ ((neiβ€˜π½)β€˜{𝐴}))))))
67 sseq2 4007 . . . . . 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 7413 . . . . . . 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 2819 . . . . . 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 631 . . . . 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 3612 . . . 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 835 . . 3 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) β†’ βˆƒπ‘” ∈ (Filβ€˜π‘‹)(𝐹 βŠ† 𝑔 ∧ 𝐴 ∈ (𝐽 fLim 𝑔)))
7372ex 413 . 2 (𝐹 ∈ (Filβ€˜π‘‹) β†’ (𝐴 ∈ (𝐽 fClus 𝐹) β†’ βˆƒπ‘” ∈ (Filβ€˜π‘‹)(𝐹 βŠ† 𝑔 ∧ 𝐴 ∈ (𝐽 fLim 𝑔))))
74 simprl 769 . . . . . 6 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝑔 ∈ (Filβ€˜π‘‹) ∧ (𝐹 βŠ† 𝑔 ∧ 𝐴 ∈ (𝐽 fLim 𝑔)))) β†’ 𝑔 ∈ (Filβ€˜π‘‹))
75 simprrr 780 . . . . . . 7 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝑔 ∈ (Filβ€˜π‘‹) ∧ (𝐹 βŠ† 𝑔 ∧ 𝐴 ∈ (𝐽 fLim 𝑔)))) β†’ 𝐴 ∈ (𝐽 fLim 𝑔))
76 flimtopon 23465 . . . . . . 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 256 . . . . 5 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝑔 ∈ (Filβ€˜π‘‹) ∧ (𝐹 βŠ† 𝑔 ∧ 𝐴 ∈ (𝐽 fLim 𝑔)))) β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
79 simpl 483 . . . . 5 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝑔 ∈ (Filβ€˜π‘‹) ∧ (𝐹 βŠ† 𝑔 ∧ 𝐴 ∈ (𝐽 fLim 𝑔)))) β†’ 𝐹 ∈ (Filβ€˜π‘‹))
80 simprrl 779 . . . . 5 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝑔 ∈ (Filβ€˜π‘‹) ∧ (𝐹 βŠ† 𝑔 ∧ 𝐴 ∈ (𝐽 fLim 𝑔)))) β†’ 𝐹 βŠ† 𝑔)
81 fclsss2 23518 . . . . 5 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐹 βŠ† 𝑔) β†’ (𝐽 fClus 𝑔) βŠ† (𝐽 fClus 𝐹))
8278, 79, 80, 81syl3anc 1371 . . . 4 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝑔 ∈ (Filβ€˜π‘‹) ∧ (𝐹 βŠ† 𝑔 ∧ 𝐴 ∈ (𝐽 fLim 𝑔)))) β†’ (𝐽 fClus 𝑔) βŠ† (𝐽 fClus 𝐹))
83 flimfcls 23521 . . . . 5 (𝐽 fLim 𝑔) βŠ† (𝐽 fClus 𝑔)
8483, 75sselid 3979 . . . 4 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝑔 ∈ (Filβ€˜π‘‹) ∧ (𝐹 βŠ† 𝑔 ∧ 𝐴 ∈ (𝐽 fLim 𝑔)))) β†’ 𝐴 ∈ (𝐽 fClus 𝑔))
8582, 84sseldd 3982 . . 3 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝑔 ∈ (Filβ€˜π‘‹) ∧ (𝐹 βŠ† 𝑔 ∧ 𝐴 ∈ (𝐽 fLim 𝑔)))) β†’ 𝐴 ∈ (𝐽 fClus 𝐹))
8685rexlimdvaa 3156 . 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 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   β‰  wne 2940  βˆ€wral 3061  βˆƒwrex 3070  Vcvv 3474   βˆͺ cun 3945   ∩ cin 3946   βŠ† wss 3947  βˆ…c0 4321  π’« cpw 4601  {csn 4627  βˆͺ cuni 4907  β€˜cfv 6540  (class class class)co 7405  ficfi 9401  fBascfbas 20924  filGencfg 20925  Topctop 22386  TopOnctopon 22403  neicnei 22592  Filcfil 23340   fLim cflim 23429   fClus cfcls 23431
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-iin 4999  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-1o 8462  df-er 8699  df-en 8936  df-fin 8939  df-fi 9402  df-fbas 20933  df-fg 20934  df-top 22387  df-topon 22404  df-cld 22514  df-ntr 22515  df-cls 22516  df-nei 22593  df-fil 23341  df-flim 23434  df-fcls 23436
This theorem is referenced by:  uffclsflim  23526  cnpfcfi  23535
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