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Theorem fclscmp 23534
Description: A space is compact iff every filter clusters. (Contributed by Jeff Hankins, 20-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)
Assertion
Ref Expression
fclscmp (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ (𝐽 ∈ Comp ↔ βˆ€π‘“ ∈ (Filβ€˜π‘‹)(𝐽 fClus 𝑓) β‰  βˆ…))
Distinct variable groups:   𝑓,𝐽   𝑓,𝑋

Proof of Theorem fclscmp
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2733 . . . . 5 βˆͺ 𝐽 = βˆͺ 𝐽
21fclscmpi 23533 . . . 4 ((𝐽 ∈ Comp ∧ 𝑓 ∈ (Filβ€˜βˆͺ 𝐽)) β†’ (𝐽 fClus 𝑓) β‰  βˆ…)
32ralrimiva 3147 . . 3 (𝐽 ∈ Comp β†’ βˆ€π‘“ ∈ (Filβ€˜βˆͺ 𝐽)(𝐽 fClus 𝑓) β‰  βˆ…)
4 toponuni 22416 . . . . 5 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 = βˆͺ 𝐽)
54fveq2d 6896 . . . 4 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ (Filβ€˜π‘‹) = (Filβ€˜βˆͺ 𝐽))
65raleqdv 3326 . . 3 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ (βˆ€π‘“ ∈ (Filβ€˜π‘‹)(𝐽 fClus 𝑓) β‰  βˆ… ↔ βˆ€π‘“ ∈ (Filβ€˜βˆͺ 𝐽)(𝐽 fClus 𝑓) β‰  βˆ…))
73, 6imbitrrid 245 . 2 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ (𝐽 ∈ Comp β†’ βˆ€π‘“ ∈ (Filβ€˜π‘‹)(𝐽 fClus 𝑓) β‰  βˆ…))
8 elpwi 4610 . . . . . 6 (π‘₯ ∈ 𝒫 (Clsdβ€˜π½) β†’ π‘₯ βŠ† (Clsdβ€˜π½))
9 vn0 4339 . . . . . . . . . 10 V β‰  βˆ…
10 simpr 486 . . . . . . . . . . . . 13 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (π‘₯ βŠ† (Clsdβ€˜π½) ∧ Β¬ βˆ… ∈ (fiβ€˜π‘₯))) ∧ π‘₯ = βˆ…) β†’ π‘₯ = βˆ…)
1110inteqd 4956 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (π‘₯ βŠ† (Clsdβ€˜π½) ∧ Β¬ βˆ… ∈ (fiβ€˜π‘₯))) ∧ π‘₯ = βˆ…) β†’ ∩ π‘₯ = ∩ βˆ…)
12 int0 4967 . . . . . . . . . . . 12 ∩ βˆ… = V
1311, 12eqtrdi 2789 . . . . . . . . . . 11 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (π‘₯ βŠ† (Clsdβ€˜π½) ∧ Β¬ βˆ… ∈ (fiβ€˜π‘₯))) ∧ π‘₯ = βˆ…) β†’ ∩ π‘₯ = V)
1413neeq1d 3001 . . . . . . . . . 10 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (π‘₯ βŠ† (Clsdβ€˜π½) ∧ Β¬ βˆ… ∈ (fiβ€˜π‘₯))) ∧ π‘₯ = βˆ…) β†’ (∩ π‘₯ β‰  βˆ… ↔ V β‰  βˆ…))
159, 14mpbiri 258 . . . . . . . . 9 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (π‘₯ βŠ† (Clsdβ€˜π½) ∧ Β¬ βˆ… ∈ (fiβ€˜π‘₯))) ∧ π‘₯ = βˆ…) β†’ ∩ π‘₯ β‰  βˆ…)
1615a1d 25 . . . . . . . 8 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (π‘₯ βŠ† (Clsdβ€˜π½) ∧ Β¬ βˆ… ∈ (fiβ€˜π‘₯))) ∧ π‘₯ = βˆ…) β†’ (βˆ€π‘“ ∈ (Filβ€˜π‘‹)(𝐽 fClus 𝑓) β‰  βˆ… β†’ ∩ π‘₯ β‰  βˆ…))
17 ssfii 9414 . . . . . . . . . . . . . . . 16 (π‘₯ ∈ V β†’ π‘₯ βŠ† (fiβ€˜π‘₯))
1817elv 3481 . . . . . . . . . . . . . . 15 π‘₯ βŠ† (fiβ€˜π‘₯)
19 simplrl 776 . . . . . . . . . . . . . . . . . 18 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (π‘₯ βŠ† (Clsdβ€˜π½) ∧ Β¬ βˆ… ∈ (fiβ€˜π‘₯))) ∧ π‘₯ β‰  βˆ…) β†’ π‘₯ βŠ† (Clsdβ€˜π½))
201cldss2 22534 . . . . . . . . . . . . . . . . . . 19 (Clsdβ€˜π½) βŠ† 𝒫 βˆͺ 𝐽
214ad2antrr 725 . . . . . . . . . . . . . . . . . . . 20 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (π‘₯ βŠ† (Clsdβ€˜π½) ∧ Β¬ βˆ… ∈ (fiβ€˜π‘₯))) ∧ π‘₯ β‰  βˆ…) β†’ 𝑋 = βˆͺ 𝐽)
2221pweqd 4620 . . . . . . . . . . . . . . . . . . 19 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (π‘₯ βŠ† (Clsdβ€˜π½) ∧ Β¬ βˆ… ∈ (fiβ€˜π‘₯))) ∧ π‘₯ β‰  βˆ…) β†’ 𝒫 𝑋 = 𝒫 βˆͺ 𝐽)
2320, 22sseqtrrid 4036 . . . . . . . . . . . . . . . . . 18 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (π‘₯ βŠ† (Clsdβ€˜π½) ∧ Β¬ βˆ… ∈ (fiβ€˜π‘₯))) ∧ π‘₯ β‰  βˆ…) β†’ (Clsdβ€˜π½) βŠ† 𝒫 𝑋)
2419, 23sstrd 3993 . . . . . . . . . . . . . . . . 17 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (π‘₯ βŠ† (Clsdβ€˜π½) ∧ Β¬ βˆ… ∈ (fiβ€˜π‘₯))) ∧ π‘₯ β‰  βˆ…) β†’ π‘₯ βŠ† 𝒫 𝑋)
25 simpr 486 . . . . . . . . . . . . . . . . 17 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (π‘₯ βŠ† (Clsdβ€˜π½) ∧ Β¬ βˆ… ∈ (fiβ€˜π‘₯))) ∧ π‘₯ β‰  βˆ…) β†’ π‘₯ β‰  βˆ…)
26 simplrr 777 . . . . . . . . . . . . . . . . 17 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (π‘₯ βŠ† (Clsdβ€˜π½) ∧ Β¬ βˆ… ∈ (fiβ€˜π‘₯))) ∧ π‘₯ β‰  βˆ…) β†’ Β¬ βˆ… ∈ (fiβ€˜π‘₯))
27 toponmax 22428 . . . . . . . . . . . . . . . . . . 19 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 ∈ 𝐽)
2827ad2antrr 725 . . . . . . . . . . . . . . . . . 18 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (π‘₯ βŠ† (Clsdβ€˜π½) ∧ Β¬ βˆ… ∈ (fiβ€˜π‘₯))) ∧ π‘₯ β‰  βˆ…) β†’ 𝑋 ∈ 𝐽)
29 fsubbas 23371 . . . . . . . . . . . . . . . . . 18 (𝑋 ∈ 𝐽 β†’ ((fiβ€˜π‘₯) ∈ (fBasβ€˜π‘‹) ↔ (π‘₯ βŠ† 𝒫 𝑋 ∧ π‘₯ β‰  βˆ… ∧ Β¬ βˆ… ∈ (fiβ€˜π‘₯))))
3028, 29syl 17 . . . . . . . . . . . . . . . . 17 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (π‘₯ βŠ† (Clsdβ€˜π½) ∧ Β¬ βˆ… ∈ (fiβ€˜π‘₯))) ∧ π‘₯ β‰  βˆ…) β†’ ((fiβ€˜π‘₯) ∈ (fBasβ€˜π‘‹) ↔ (π‘₯ βŠ† 𝒫 𝑋 ∧ π‘₯ β‰  βˆ… ∧ Β¬ βˆ… ∈ (fiβ€˜π‘₯))))
3124, 25, 26, 30mpbir3and 1343 . . . . . . . . . . . . . . . 16 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (π‘₯ βŠ† (Clsdβ€˜π½) ∧ Β¬ βˆ… ∈ (fiβ€˜π‘₯))) ∧ π‘₯ β‰  βˆ…) β†’ (fiβ€˜π‘₯) ∈ (fBasβ€˜π‘‹))
32 ssfg 23376 . . . . . . . . . . . . . . . 16 ((fiβ€˜π‘₯) ∈ (fBasβ€˜π‘‹) β†’ (fiβ€˜π‘₯) βŠ† (𝑋filGen(fiβ€˜π‘₯)))
3331, 32syl 17 . . . . . . . . . . . . . . 15 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (π‘₯ βŠ† (Clsdβ€˜π½) ∧ Β¬ βˆ… ∈ (fiβ€˜π‘₯))) ∧ π‘₯ β‰  βˆ…) β†’ (fiβ€˜π‘₯) βŠ† (𝑋filGen(fiβ€˜π‘₯)))
3418, 33sstrid 3994 . . . . . . . . . . . . . 14 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (π‘₯ βŠ† (Clsdβ€˜π½) ∧ Β¬ βˆ… ∈ (fiβ€˜π‘₯))) ∧ π‘₯ β‰  βˆ…) β†’ π‘₯ βŠ† (𝑋filGen(fiβ€˜π‘₯)))
3534sselda 3983 . . . . . . . . . . . . 13 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (π‘₯ βŠ† (Clsdβ€˜π½) ∧ Β¬ βˆ… ∈ (fiβ€˜π‘₯))) ∧ π‘₯ β‰  βˆ…) ∧ 𝑦 ∈ π‘₯) β†’ 𝑦 ∈ (𝑋filGen(fiβ€˜π‘₯)))
36 fclssscls 23522 . . . . . . . . . . . . 13 (𝑦 ∈ (𝑋filGen(fiβ€˜π‘₯)) β†’ (𝐽 fClus (𝑋filGen(fiβ€˜π‘₯))) βŠ† ((clsβ€˜π½)β€˜π‘¦))
3735, 36syl 17 . . . . . . . . . . . 12 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (π‘₯ βŠ† (Clsdβ€˜π½) ∧ Β¬ βˆ… ∈ (fiβ€˜π‘₯))) ∧ π‘₯ β‰  βˆ…) ∧ 𝑦 ∈ π‘₯) β†’ (𝐽 fClus (𝑋filGen(fiβ€˜π‘₯))) βŠ† ((clsβ€˜π½)β€˜π‘¦))
3819sselda 3983 . . . . . . . . . . . . 13 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (π‘₯ βŠ† (Clsdβ€˜π½) ∧ Β¬ βˆ… ∈ (fiβ€˜π‘₯))) ∧ π‘₯ β‰  βˆ…) ∧ 𝑦 ∈ π‘₯) β†’ 𝑦 ∈ (Clsdβ€˜π½))
39 cldcls 22546 . . . . . . . . . . . . 13 (𝑦 ∈ (Clsdβ€˜π½) β†’ ((clsβ€˜π½)β€˜π‘¦) = 𝑦)
4038, 39syl 17 . . . . . . . . . . . 12 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (π‘₯ βŠ† (Clsdβ€˜π½) ∧ Β¬ βˆ… ∈ (fiβ€˜π‘₯))) ∧ π‘₯ β‰  βˆ…) ∧ 𝑦 ∈ π‘₯) β†’ ((clsβ€˜π½)β€˜π‘¦) = 𝑦)
4137, 40sseqtrd 4023 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (π‘₯ βŠ† (Clsdβ€˜π½) ∧ Β¬ βˆ… ∈ (fiβ€˜π‘₯))) ∧ π‘₯ β‰  βˆ…) ∧ 𝑦 ∈ π‘₯) β†’ (𝐽 fClus (𝑋filGen(fiβ€˜π‘₯))) βŠ† 𝑦)
4241ralrimiva 3147 . . . . . . . . . 10 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (π‘₯ βŠ† (Clsdβ€˜π½) ∧ Β¬ βˆ… ∈ (fiβ€˜π‘₯))) ∧ π‘₯ β‰  βˆ…) β†’ βˆ€π‘¦ ∈ π‘₯ (𝐽 fClus (𝑋filGen(fiβ€˜π‘₯))) βŠ† 𝑦)
43 ssint 4969 . . . . . . . . . 10 ((𝐽 fClus (𝑋filGen(fiβ€˜π‘₯))) βŠ† ∩ π‘₯ ↔ βˆ€π‘¦ ∈ π‘₯ (𝐽 fClus (𝑋filGen(fiβ€˜π‘₯))) βŠ† 𝑦)
4442, 43sylibr 233 . . . . . . . . 9 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (π‘₯ βŠ† (Clsdβ€˜π½) ∧ Β¬ βˆ… ∈ (fiβ€˜π‘₯))) ∧ π‘₯ β‰  βˆ…) β†’ (𝐽 fClus (𝑋filGen(fiβ€˜π‘₯))) βŠ† ∩ π‘₯)
45 fgcl 23382 . . . . . . . . . 10 ((fiβ€˜π‘₯) ∈ (fBasβ€˜π‘‹) β†’ (𝑋filGen(fiβ€˜π‘₯)) ∈ (Filβ€˜π‘‹))
46 oveq2 7417 . . . . . . . . . . . 12 (𝑓 = (𝑋filGen(fiβ€˜π‘₯)) β†’ (𝐽 fClus 𝑓) = (𝐽 fClus (𝑋filGen(fiβ€˜π‘₯))))
4746neeq1d 3001 . . . . . . . . . . 11 (𝑓 = (𝑋filGen(fiβ€˜π‘₯)) β†’ ((𝐽 fClus 𝑓) β‰  βˆ… ↔ (𝐽 fClus (𝑋filGen(fiβ€˜π‘₯))) β‰  βˆ…))
4847rspcv 3609 . . . . . . . . . 10 ((𝑋filGen(fiβ€˜π‘₯)) ∈ (Filβ€˜π‘‹) β†’ (βˆ€π‘“ ∈ (Filβ€˜π‘‹)(𝐽 fClus 𝑓) β‰  βˆ… β†’ (𝐽 fClus (𝑋filGen(fiβ€˜π‘₯))) β‰  βˆ…))
4931, 45, 483syl 18 . . . . . . . . 9 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (π‘₯ βŠ† (Clsdβ€˜π½) ∧ Β¬ βˆ… ∈ (fiβ€˜π‘₯))) ∧ π‘₯ β‰  βˆ…) β†’ (βˆ€π‘“ ∈ (Filβ€˜π‘‹)(𝐽 fClus 𝑓) β‰  βˆ… β†’ (𝐽 fClus (𝑋filGen(fiβ€˜π‘₯))) β‰  βˆ…))
50 ssn0 4401 . . . . . . . . 9 (((𝐽 fClus (𝑋filGen(fiβ€˜π‘₯))) βŠ† ∩ π‘₯ ∧ (𝐽 fClus (𝑋filGen(fiβ€˜π‘₯))) β‰  βˆ…) β†’ ∩ π‘₯ β‰  βˆ…)
5144, 49, 50syl6an 683 . . . . . . . 8 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (π‘₯ βŠ† (Clsdβ€˜π½) ∧ Β¬ βˆ… ∈ (fiβ€˜π‘₯))) ∧ π‘₯ β‰  βˆ…) β†’ (βˆ€π‘“ ∈ (Filβ€˜π‘‹)(𝐽 fClus 𝑓) β‰  βˆ… β†’ ∩ π‘₯ β‰  βˆ…))
5216, 51pm2.61dane 3030 . . . . . . 7 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (π‘₯ βŠ† (Clsdβ€˜π½) ∧ Β¬ βˆ… ∈ (fiβ€˜π‘₯))) β†’ (βˆ€π‘“ ∈ (Filβ€˜π‘‹)(𝐽 fClus 𝑓) β‰  βˆ… β†’ ∩ π‘₯ β‰  βˆ…))
5352expr 458 . . . . . 6 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘₯ βŠ† (Clsdβ€˜π½)) β†’ (Β¬ βˆ… ∈ (fiβ€˜π‘₯) β†’ (βˆ€π‘“ ∈ (Filβ€˜π‘‹)(𝐽 fClus 𝑓) β‰  βˆ… β†’ ∩ π‘₯ β‰  βˆ…)))
548, 53sylan2 594 . . . . 5 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘₯ ∈ 𝒫 (Clsdβ€˜π½)) β†’ (Β¬ βˆ… ∈ (fiβ€˜π‘₯) β†’ (βˆ€π‘“ ∈ (Filβ€˜π‘‹)(𝐽 fClus 𝑓) β‰  βˆ… β†’ ∩ π‘₯ β‰  βˆ…)))
5554com23 86 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘₯ ∈ 𝒫 (Clsdβ€˜π½)) β†’ (βˆ€π‘“ ∈ (Filβ€˜π‘‹)(𝐽 fClus 𝑓) β‰  βˆ… β†’ (Β¬ βˆ… ∈ (fiβ€˜π‘₯) β†’ ∩ π‘₯ β‰  βˆ…)))
5655ralrimdva 3155 . . 3 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ (βˆ€π‘“ ∈ (Filβ€˜π‘‹)(𝐽 fClus 𝑓) β‰  βˆ… β†’ βˆ€π‘₯ ∈ 𝒫 (Clsdβ€˜π½)(Β¬ βˆ… ∈ (fiβ€˜π‘₯) β†’ ∩ π‘₯ β‰  βˆ…)))
57 topontop 22415 . . . 4 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝐽 ∈ Top)
58 cmpfi 22912 . . . 4 (𝐽 ∈ Top β†’ (𝐽 ∈ Comp ↔ βˆ€π‘₯ ∈ 𝒫 (Clsdβ€˜π½)(Β¬ βˆ… ∈ (fiβ€˜π‘₯) β†’ ∩ π‘₯ β‰  βˆ…)))
5957, 58syl 17 . . 3 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ (𝐽 ∈ Comp ↔ βˆ€π‘₯ ∈ 𝒫 (Clsdβ€˜π½)(Β¬ βˆ… ∈ (fiβ€˜π‘₯) β†’ ∩ π‘₯ β‰  βˆ…)))
6056, 59sylibrd 259 . 2 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ (βˆ€π‘“ ∈ (Filβ€˜π‘‹)(𝐽 fClus 𝑓) β‰  βˆ… β†’ 𝐽 ∈ Comp))
617, 60impbid 211 1 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ (𝐽 ∈ Comp ↔ βˆ€π‘“ ∈ (Filβ€˜π‘‹)(𝐽 fClus 𝑓) β‰  βˆ…))
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  Vcvv 3475   βŠ† wss 3949  βˆ…c0 4323  π’« cpw 4603  βˆͺ cuni 4909  βˆ© cint 4951  β€˜cfv 6544  (class class class)co 7409  ficfi 9405  fBascfbas 20932  filGencfg 20933  Topctop 22395  TopOnctopon 22412  Clsdccld 22520  clsccl 22522  Compccmp 22890  Filcfil 23349   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-cls 22525  df-cmp 22891  df-fil 23350  df-fcls 23445
This theorem is referenced by:  ufilcmp  23536
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