MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fclscmp Structured version   Visualization version   GIF version

Theorem fclscmp 23404
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 23403 . . . 4 ((𝐽 ∈ Comp ∧ 𝑓 ∈ (Filβ€˜βˆͺ 𝐽)) β†’ (𝐽 fClus 𝑓) β‰  βˆ…)
32ralrimiva 3140 . . 3 (𝐽 ∈ Comp β†’ βˆ€π‘“ ∈ (Filβ€˜βˆͺ 𝐽)(𝐽 fClus 𝑓) β‰  βˆ…)
4 toponuni 22286 . . . . 5 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 = βˆͺ 𝐽)
54fveq2d 6850 . . . 4 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ (Filβ€˜π‘‹) = (Filβ€˜βˆͺ 𝐽))
65raleqdv 3312 . . 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, 6syl5ibr 246 . 2 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ (𝐽 ∈ Comp β†’ βˆ€π‘“ ∈ (Filβ€˜π‘‹)(𝐽 fClus 𝑓) β‰  βˆ…))
8 elpwi 4571 . . . . . 6 (π‘₯ ∈ 𝒫 (Clsdβ€˜π½) β†’ π‘₯ βŠ† (Clsdβ€˜π½))
9 vn0 4302 . . . . . . . . . 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 4916 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (π‘₯ βŠ† (Clsdβ€˜π½) ∧ Β¬ βˆ… ∈ (fiβ€˜π‘₯))) ∧ π‘₯ = βˆ…) β†’ ∩ π‘₯ = ∩ βˆ…)
12 int0 4927 . . . . . . . . . . . 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 3000 . . . . . . . . . 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 9363 . . . . . . . . . . . . . . . 16 (π‘₯ ∈ V β†’ π‘₯ βŠ† (fiβ€˜π‘₯))
1817elv 3453 . . . . . . . . . . . . . . 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 22404 . . . . . . . . . . . . . . . . . . 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 4581 . . . . . . . . . . . . . . . . . . 19 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (π‘₯ βŠ† (Clsdβ€˜π½) ∧ Β¬ βˆ… ∈ (fiβ€˜π‘₯))) ∧ π‘₯ β‰  βˆ…) β†’ 𝒫 𝑋 = 𝒫 βˆͺ 𝐽)
2320, 22sseqtrrid 4001 . . . . . . . . . . . . . . . . . 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 3958 . . . . . . . . . . . . . . . . 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 22298 . . . . . . . . . . . . . . . . . . 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 23241 . . . . . . . . . . . . . . . . . 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 23246 . . . . . . . . . . . . . . . 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 3959 . . . . . . . . . . . . . 14 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (π‘₯ βŠ† (Clsdβ€˜π½) ∧ Β¬ βˆ… ∈ (fiβ€˜π‘₯))) ∧ π‘₯ β‰  βˆ…) β†’ π‘₯ βŠ† (𝑋filGen(fiβ€˜π‘₯)))
3534sselda 3948 . . . . . . . . . . . . 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 23392 . . . . . . . . . . . . 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 3948 . . . . . . . . . . . . 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 22416 . . . . . . . . . . . . 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 3988 . . . . . . . . . . 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 3140 . . . . . . . . . 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 4929 . . . . . . . . . 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 23252 . . . . . . . . . 10 ((fiβ€˜π‘₯) ∈ (fBasβ€˜π‘‹) β†’ (𝑋filGen(fiβ€˜π‘₯)) ∈ (Filβ€˜π‘‹))
46 oveq2 7369 . . . . . . . . . . . 12 (𝑓 = (𝑋filGen(fiβ€˜π‘₯)) β†’ (𝐽 fClus 𝑓) = (𝐽 fClus (𝑋filGen(fiβ€˜π‘₯))))
4746neeq1d 3000 . . . . . . . . . . 11 (𝑓 = (𝑋filGen(fiβ€˜π‘₯)) β†’ ((𝐽 fClus 𝑓) β‰  βˆ… ↔ (𝐽 fClus (𝑋filGen(fiβ€˜π‘₯))) β‰  βˆ…))
4847rspcv 3579 . . . . . . . . . 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 4364 . . . . . . . . 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 3029 . . . . . . 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 3148 . . 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 22285 . . . 4 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝐽 ∈ Top)
58 cmpfi 22782 . . . 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 2940  βˆ€wral 3061  Vcvv 3447   βŠ† wss 3914  βˆ…c0 4286  π’« cpw 4564  βˆͺ cuni 4869  βˆ© cint 4911  β€˜cfv 6500  (class class class)co 7361  ficfi 9354  fBascfbas 20807  filGencfg 20808  Topctop 22265  TopOnctopon 22282  Clsdccld 22390  clsccl 22392  Compccmp 22760  Filcfil 23219   fClus cfcls 23310
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 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
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 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-int 4912  df-iun 4960  df-iin 4961  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-1o 8416  df-er 8654  df-en 8890  df-fin 8893  df-fi 9355  df-fbas 20816  df-fg 20817  df-top 22266  df-topon 22283  df-cld 22393  df-cls 22395  df-cmp 22761  df-fil 23220  df-fcls 23315
This theorem is referenced by:  ufilcmp  23406
  Copyright terms: Public domain W3C validator