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Theorem fclsfnflim 23751
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 23575 . . . . . . . 8 (𝐹 ∈ (Filβ€˜π‘‹) β†’ 𝐹 βŠ† 𝒫 𝑋)
21adantr 479 . . . . . . 7 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) β†’ 𝐹 βŠ† 𝒫 𝑋)
3 fclstop 23735 . . . . . . . . . 10 (𝐴 ∈ (𝐽 fClus 𝐹) β†’ 𝐽 ∈ Top)
43adantl 480 . . . . . . . . 9 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) β†’ 𝐽 ∈ Top)
5 eqid 2730 . . . . . . . . . 10 βˆͺ 𝐽 = βˆͺ 𝐽
65neisspw 22831 . . . . . . . . 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 23605 . . . . . . . . . 10 (𝐹 ∈ (Filβ€˜π‘‹) β†’ βˆͺ 𝐹 = 𝑋)
95fclsfil 23734 . . . . . . . . . . 11 (𝐴 ∈ (𝐽 fClus 𝐹) β†’ 𝐹 ∈ (Filβ€˜βˆͺ 𝐽))
10 filunibas 23605 . . . . . . . . . . 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 2791 . . . . . . . . 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 23586 . . . . . . . 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 585 . . . . . . 7 (𝐹 ∈ (Filβ€˜π‘‹) β†’ (𝐹 βˆͺ ((neiβ€˜π½)β€˜{𝐴})) β‰  βˆ…)
2019adantr 479 . . . . . 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 23741 . . . . . . . . . . . 12 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑦 ∈ ((neiβ€˜π½)β€˜{𝐴}) ∧ π‘₯ ∈ 𝐹) β†’ (𝑦 ∩ π‘₯) β‰  βˆ…)
2321, 22eqnetrrid 3014 . . . . . . . . . . 11 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑦 ∈ ((neiβ€˜π½)β€˜{𝐴}) ∧ π‘₯ ∈ 𝐹) β†’ (π‘₯ ∩ 𝑦) β‰  βˆ…)
24233com23 1124 . . . . . . . . . 10 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ π‘₯ ∈ 𝐹 ∧ 𝑦 ∈ ((neiβ€˜π½)β€˜{𝐴})) β†’ (π‘₯ ∩ 𝑦) β‰  βˆ…)
25243expb 1118 . . . . . . . . 9 ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ (π‘₯ ∈ 𝐹 ∧ 𝑦 ∈ ((neiβ€˜π½)β€˜{𝐴}))) β†’ (π‘₯ ∩ 𝑦) β‰  βˆ…)
2625adantll 710 . . . . . . . 8 (((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) ∧ (π‘₯ ∈ 𝐹 ∧ 𝑦 ∈ ((neiβ€˜π½)β€˜{𝐴}))) β†’ (π‘₯ ∩ 𝑦) β‰  βˆ…)
2726ralrimivva 3198 . . . . . . 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 23572 . . . . . . . . 9 (𝐹 ∈ (Filβ€˜π‘‹) β†’ 𝐹 ∈ (fBasβ€˜π‘‹))
2928adantr 479 . . . . . . . 8 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) β†’ 𝐹 ∈ (fBasβ€˜π‘‹))
30 istopon 22634 . . . . . . . . . . 11 (𝐽 ∈ (TopOnβ€˜π‘‹) ↔ (𝐽 ∈ Top ∧ 𝑋 = βˆͺ 𝐽))
314, 12, 30sylanbrc 581 . . . . . . . . . 10 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
325fclselbas 23740 . . . . . . . . . . . . 13 (𝐴 ∈ (𝐽 fClus 𝐹) β†’ 𝐴 ∈ βˆͺ 𝐽)
3332adantl 480 . . . . . . . . . . . 12 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) β†’ 𝐴 ∈ βˆͺ 𝐽)
3433, 12eleqtrrd 2834 . . . . . . . . . . 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 480 . . . . . . . . . 10 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) β†’ {𝐴} β‰  βˆ…)
38 neifil 23604 . . . . . . . . . 10 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ {𝐴} βŠ† 𝑋 ∧ {𝐴} β‰  βˆ…) β†’ ((neiβ€˜π½)β€˜{𝐴}) ∈ (Filβ€˜π‘‹))
3931, 35, 37, 38syl3anc 1369 . . . . . . . . 9 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) β†’ ((neiβ€˜π½)β€˜{𝐴}) ∈ (Filβ€˜π‘‹))
40 filfbas 23572 . . . . . . . . 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 23593 . . . . . . . 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 582 . . . . . . 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 23579 . . . . . . . 8 (𝐹 ∈ (Filβ€˜π‘‹) β†’ 𝑋 ∈ 𝐹)
46 fsubbas 23591 . . . . . . . 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 479 . . . . . 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 1340 . . . . 5 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) β†’ (fiβ€˜(𝐹 βˆͺ ((neiβ€˜π½)β€˜{𝐴}))) ∈ (fBasβ€˜π‘‹))
50 fgcl 23602 . . . . 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 6903 . . . . . . . . 9 ((neiβ€˜π½)β€˜{𝐴}) ∈ V
53 unexg 7738 . . . . . . . . 9 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ ((neiβ€˜π½)β€˜{𝐴}) ∈ V) β†’ (𝐹 βˆͺ ((neiβ€˜π½)β€˜{𝐴})) ∈ V)
5452, 53mpan2 687 . . . . . . . 8 (𝐹 ∈ (Filβ€˜π‘‹) β†’ (𝐹 βˆͺ ((neiβ€˜π½)β€˜{𝐴})) ∈ V)
55 ssfii 9416 . . . . . . . 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 479 . . . . . 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 23596 . . . . . 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 23695 . . . . . 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 582 . . . . 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 709 . . . 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 7419 . . . . . . 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 2817 . . . . . 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 629 . . . . 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 3611 . . . 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 833 . . 3 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐴 ∈ (𝐽 fClus 𝐹)) β†’ βˆƒπ‘” ∈ (Filβ€˜π‘‹)(𝐹 βŠ† 𝑔 ∧ 𝐴 ∈ (𝐽 fLim 𝑔)))
7372ex 411 . 2 (𝐹 ∈ (Filβ€˜π‘‹) β†’ (𝐴 ∈ (𝐽 fClus 𝐹) β†’ βˆƒπ‘” ∈ (Filβ€˜π‘‹)(𝐹 βŠ† 𝑔 ∧ 𝐴 ∈ (𝐽 fLim 𝑔))))
74 simprl 767 . . . . . 6 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝑔 ∈ (Filβ€˜π‘‹) ∧ (𝐹 βŠ† 𝑔 ∧ 𝐴 ∈ (𝐽 fLim 𝑔)))) β†’ 𝑔 ∈ (Filβ€˜π‘‹))
75 simprrr 778 . . . . . . 7 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝑔 ∈ (Filβ€˜π‘‹) ∧ (𝐹 βŠ† 𝑔 ∧ 𝐴 ∈ (𝐽 fLim 𝑔)))) β†’ 𝐴 ∈ (𝐽 fLim 𝑔))
76 flimtopon 23694 . . . . . . 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 481 . . . . 5 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝑔 ∈ (Filβ€˜π‘‹) ∧ (𝐹 βŠ† 𝑔 ∧ 𝐴 ∈ (𝐽 fLim 𝑔)))) β†’ 𝐹 ∈ (Filβ€˜π‘‹))
80 simprrl 777 . . . . 5 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝑔 ∈ (Filβ€˜π‘‹) ∧ (𝐹 βŠ† 𝑔 ∧ 𝐴 ∈ (𝐽 fLim 𝑔)))) β†’ 𝐹 βŠ† 𝑔)
81 fclsss2 23747 . . . . 5 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐹 βŠ† 𝑔) β†’ (𝐽 fClus 𝑔) βŠ† (𝐽 fClus 𝐹))
8278, 79, 80, 81syl3anc 1369 . . . 4 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ (𝑔 ∈ (Filβ€˜π‘‹) ∧ (𝐹 βŠ† 𝑔 ∧ 𝐴 ∈ (𝐽 fLim 𝑔)))) β†’ (𝐽 fClus 𝑔) βŠ† (𝐽 fClus 𝐹))
83 flimfcls 23750 . . . . 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 3154 . 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 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104   β‰  wne 2938  βˆ€wral 3059  βˆƒwrex 3068  Vcvv 3472   βˆͺ cun 3945   ∩ cin 3946   βŠ† wss 3947  βˆ…c0 4321  π’« cpw 4601  {csn 4627  βˆͺ cuni 4907  β€˜cfv 6542  (class class class)co 7411  ficfi 9407  fBascfbas 21132  filGencfg 21133  Topctop 22615  TopOnctopon 22632  neicnei 22821  Filcfil 23569   fLim cflim 23658   fClus cfcls 23660
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  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 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1o 8468  df-er 8705  df-en 8942  df-fin 8945  df-fi 9408  df-fbas 21141  df-fg 21142  df-top 22616  df-topon 22633  df-cld 22743  df-ntr 22744  df-cls 22745  df-nei 22822  df-fil 23570  df-flim 23663  df-fcls 23665
This theorem is referenced by:  uffclsflim  23755  cnpfcfi  23764
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