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Theorem fclscmp 22054
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 ∈ (TopOn‘𝑋) → (𝐽 ∈ Comp ↔ ∀𝑓 ∈ (Fil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅))
Distinct variable groups:   𝑓,𝐽   𝑓,𝑋

Proof of Theorem fclscmp
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2771 . . . . 5 𝐽 = 𝐽
21fclscmpi 22053 . . . 4 ((𝐽 ∈ Comp ∧ 𝑓 ∈ (Fil‘ 𝐽)) → (𝐽 fClus 𝑓) ≠ ∅)
32ralrimiva 3115 . . 3 (𝐽 ∈ Comp → ∀𝑓 ∈ (Fil‘ 𝐽)(𝐽 fClus 𝑓) ≠ ∅)
4 toponuni 20939 . . . . 5 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
54fveq2d 6336 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → (Fil‘𝑋) = (Fil‘ 𝐽))
65raleqdv 3293 . . 3 (𝐽 ∈ (TopOn‘𝑋) → (∀𝑓 ∈ (Fil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ ↔ ∀𝑓 ∈ (Fil‘ 𝐽)(𝐽 fClus 𝑓) ≠ ∅))
73, 6syl5ibr 236 . 2 (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Comp → ∀𝑓 ∈ (Fil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅))
8 elpwi 4307 . . . . . 6 (𝑥 ∈ 𝒫 (Clsd‘𝐽) → 𝑥 ⊆ (Clsd‘𝐽))
9 vn0 4072 . . . . . . . . . 10 V ≠ ∅
10 simpr 471 . . . . . . . . . . . . 13 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 = ∅) → 𝑥 = ∅)
1110inteqd 4616 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 = ∅) → 𝑥 = ∅)
12 int0 4625 . . . . . . . . . . . 12 ∅ = V
1311, 12syl6eq 2821 . . . . . . . . . . 11 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 = ∅) → 𝑥 = V)
1413neeq1d 3002 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 = ∅) → ( 𝑥 ≠ ∅ ↔ V ≠ ∅))
159, 14mpbiri 248 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 = ∅) → 𝑥 ≠ ∅)
1615a1d 25 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 = ∅) → (∀𝑓 ∈ (Fil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ → 𝑥 ≠ ∅))
17 vex 3354 . . . . . . . . . . . . . . . 16 𝑥 ∈ V
18 ssfii 8481 . . . . . . . . . . . . . . . 16 (𝑥 ∈ V → 𝑥 ⊆ (fi‘𝑥))
1917, 18ax-mp 5 . . . . . . . . . . . . . . 15 𝑥 ⊆ (fi‘𝑥)
20 simplrl 754 . . . . . . . . . . . . . . . . . 18 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) → 𝑥 ⊆ (Clsd‘𝐽))
211cldss2 21055 . . . . . . . . . . . . . . . . . . 19 (Clsd‘𝐽) ⊆ 𝒫 𝐽
224ad2antrr 697 . . . . . . . . . . . . . . . . . . . 20 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) → 𝑋 = 𝐽)
2322pweqd 4302 . . . . . . . . . . . . . . . . . . 19 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) → 𝒫 𝑋 = 𝒫 𝐽)
2421, 23syl5sseqr 3803 . . . . . . . . . . . . . . . . . 18 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) → (Clsd‘𝐽) ⊆ 𝒫 𝑋)
2520, 24sstrd 3762 . . . . . . . . . . . . . . . . 17 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) → 𝑥 ⊆ 𝒫 𝑋)
26 simpr 471 . . . . . . . . . . . . . . . . 17 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) → 𝑥 ≠ ∅)
27 simplrr 755 . . . . . . . . . . . . . . . . 17 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) → ¬ ∅ ∈ (fi‘𝑥))
28 toponmax 20951 . . . . . . . . . . . . . . . . . . 19 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
2928ad2antrr 697 . . . . . . . . . . . . . . . . . 18 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) → 𝑋𝐽)
30 fsubbas 21891 . . . . . . . . . . . . . . . . . 18 (𝑋𝐽 → ((fi‘𝑥) ∈ (fBas‘𝑋) ↔ (𝑥 ⊆ 𝒫 𝑋𝑥 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝑥))))
3129, 30syl 17 . . . . . . . . . . . . . . . . 17 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) → ((fi‘𝑥) ∈ (fBas‘𝑋) ↔ (𝑥 ⊆ 𝒫 𝑋𝑥 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝑥))))
3225, 26, 27, 31mpbir3and 1427 . . . . . . . . . . . . . . . 16 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) → (fi‘𝑥) ∈ (fBas‘𝑋))
33 ssfg 21896 . . . . . . . . . . . . . . . 16 ((fi‘𝑥) ∈ (fBas‘𝑋) → (fi‘𝑥) ⊆ (𝑋filGen(fi‘𝑥)))
3432, 33syl 17 . . . . . . . . . . . . . . 15 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) → (fi‘𝑥) ⊆ (𝑋filGen(fi‘𝑥)))
3519, 34syl5ss 3763 . . . . . . . . . . . . . 14 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) → 𝑥 ⊆ (𝑋filGen(fi‘𝑥)))
3635sselda 3752 . . . . . . . . . . . . 13 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) ∧ 𝑦𝑥) → 𝑦 ∈ (𝑋filGen(fi‘𝑥)))
37 fclssscls 22042 . . . . . . . . . . . . 13 (𝑦 ∈ (𝑋filGen(fi‘𝑥)) → (𝐽 fClus (𝑋filGen(fi‘𝑥))) ⊆ ((cls‘𝐽)‘𝑦))
3836, 37syl 17 . . . . . . . . . . . 12 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) ∧ 𝑦𝑥) → (𝐽 fClus (𝑋filGen(fi‘𝑥))) ⊆ ((cls‘𝐽)‘𝑦))
3920sselda 3752 . . . . . . . . . . . . 13 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) ∧ 𝑦𝑥) → 𝑦 ∈ (Clsd‘𝐽))
40 cldcls 21067 . . . . . . . . . . . . 13 (𝑦 ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘𝑦) = 𝑦)
4139, 40syl 17 . . . . . . . . . . . 12 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) ∧ 𝑦𝑥) → ((cls‘𝐽)‘𝑦) = 𝑦)
4238, 41sseqtrd 3790 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) ∧ 𝑦𝑥) → (𝐽 fClus (𝑋filGen(fi‘𝑥))) ⊆ 𝑦)
4342ralrimiva 3115 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) → ∀𝑦𝑥 (𝐽 fClus (𝑋filGen(fi‘𝑥))) ⊆ 𝑦)
44 ssint 4627 . . . . . . . . . 10 ((𝐽 fClus (𝑋filGen(fi‘𝑥))) ⊆ 𝑥 ↔ ∀𝑦𝑥 (𝐽 fClus (𝑋filGen(fi‘𝑥))) ⊆ 𝑦)
4543, 44sylibr 224 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) → (𝐽 fClus (𝑋filGen(fi‘𝑥))) ⊆ 𝑥)
46 fgcl 21902 . . . . . . . . . 10 ((fi‘𝑥) ∈ (fBas‘𝑋) → (𝑋filGen(fi‘𝑥)) ∈ (Fil‘𝑋))
47 oveq2 6801 . . . . . . . . . . . 12 (𝑓 = (𝑋filGen(fi‘𝑥)) → (𝐽 fClus 𝑓) = (𝐽 fClus (𝑋filGen(fi‘𝑥))))
4847neeq1d 3002 . . . . . . . . . . 11 (𝑓 = (𝑋filGen(fi‘𝑥)) → ((𝐽 fClus 𝑓) ≠ ∅ ↔ (𝐽 fClus (𝑋filGen(fi‘𝑥))) ≠ ∅))
4948rspcv 3456 . . . . . . . . . 10 ((𝑋filGen(fi‘𝑥)) ∈ (Fil‘𝑋) → (∀𝑓 ∈ (Fil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ → (𝐽 fClus (𝑋filGen(fi‘𝑥))) ≠ ∅))
5032, 46, 493syl 18 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) → (∀𝑓 ∈ (Fil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ → (𝐽 fClus (𝑋filGen(fi‘𝑥))) ≠ ∅))
51 ssn0 4120 . . . . . . . . 9 (((𝐽 fClus (𝑋filGen(fi‘𝑥))) ⊆ 𝑥 ∧ (𝐽 fClus (𝑋filGen(fi‘𝑥))) ≠ ∅) → 𝑥 ≠ ∅)
5245, 50, 51syl6an 655 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) → (∀𝑓 ∈ (Fil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ → 𝑥 ≠ ∅))
5316, 52pm2.61dane 3030 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) → (∀𝑓 ∈ (Fil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ → 𝑥 ≠ ∅))
5453expr 444 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ (Clsd‘𝐽)) → (¬ ∅ ∈ (fi‘𝑥) → (∀𝑓 ∈ (Fil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ → 𝑥 ≠ ∅)))
558, 54sylan2 572 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) → (¬ ∅ ∈ (fi‘𝑥) → (∀𝑓 ∈ (Fil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ → 𝑥 ≠ ∅)))
5655com23 86 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) → (∀𝑓 ∈ (Fil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ → (¬ ∅ ∈ (fi‘𝑥) → 𝑥 ≠ ∅)))
5756ralrimdva 3118 . . 3 (𝐽 ∈ (TopOn‘𝑋) → (∀𝑓 ∈ (Fil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ → ∀𝑥 ∈ 𝒫 (Clsd‘𝐽)(¬ ∅ ∈ (fi‘𝑥) → 𝑥 ≠ ∅)))
58 topontop 20938 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
59 cmpfi 21432 . . . 4 (𝐽 ∈ Top → (𝐽 ∈ Comp ↔ ∀𝑥 ∈ 𝒫 (Clsd‘𝐽)(¬ ∅ ∈ (fi‘𝑥) → 𝑥 ≠ ∅)))
6058, 59syl 17 . . 3 (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Comp ↔ ∀𝑥 ∈ 𝒫 (Clsd‘𝐽)(¬ ∅ ∈ (fi‘𝑥) → 𝑥 ≠ ∅)))
6157, 60sylibrd 249 . 2 (𝐽 ∈ (TopOn‘𝑋) → (∀𝑓 ∈ (Fil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ → 𝐽 ∈ Comp))
627, 61impbid 202 1 (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Comp ↔ ∀𝑓 ∈ (Fil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  w3a 1071   = wceq 1631  wcel 2145  wne 2943  wral 3061  Vcvv 3351  wss 3723  c0 4063  𝒫 cpw 4297   cuni 4574   cint 4611  cfv 6031  (class class class)co 6793  ficfi 8472  fBascfbas 19949  filGencfg 19950  Topctop 20918  TopOnctopon 20935  Clsdccld 21041  clsccl 21043  Compccmp 21410  Filcfil 21869   fClus cfcls 21960
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-iin 4657  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-2o 7714  df-oadd 7717  df-er 7896  df-map 8011  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-fi 8473  df-fbas 19958  df-fg 19959  df-top 20919  df-topon 20936  df-cld 21044  df-cls 21046  df-cmp 21411  df-fil 21870  df-fcls 21965
This theorem is referenced by:  ufilcmp  22056
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