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Theorem fclscmp 22630
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 2819 . . . . 5 𝐽 = 𝐽
21fclscmpi 22629 . . . 4 ((𝐽 ∈ Comp ∧ 𝑓 ∈ (Fil‘ 𝐽)) → (𝐽 fClus 𝑓) ≠ ∅)
32ralrimiva 3180 . . 3 (𝐽 ∈ Comp → ∀𝑓 ∈ (Fil‘ 𝐽)(𝐽 fClus 𝑓) ≠ ∅)
4 toponuni 21514 . . . . 5 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
54fveq2d 6667 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → (Fil‘𝑋) = (Fil‘ 𝐽))
65raleqdv 3414 . . 3 (𝐽 ∈ (TopOn‘𝑋) → (∀𝑓 ∈ (Fil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ ↔ ∀𝑓 ∈ (Fil‘ 𝐽)(𝐽 fClus 𝑓) ≠ ∅))
73, 6syl5ibr 248 . 2 (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Comp → ∀𝑓 ∈ (Fil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅))
8 elpwi 4549 . . . . . 6 (𝑥 ∈ 𝒫 (Clsd‘𝐽) → 𝑥 ⊆ (Clsd‘𝐽))
9 vn0 4302 . . . . . . . . . 10 V ≠ ∅
10 simpr 487 . . . . . . . . . . . . 13 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 = ∅) → 𝑥 = ∅)
1110inteqd 4872 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 = ∅) → 𝑥 = ∅)
12 int0 4881 . . . . . . . . . . . 12 ∅ = V
1311, 12syl6eq 2870 . . . . . . . . . . 11 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 = ∅) → 𝑥 = V)
1413neeq1d 3073 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 = ∅) → ( 𝑥 ≠ ∅ ↔ V ≠ ∅))
159, 14mpbiri 260 . . . . . . . . 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 ssfii 8875 . . . . . . . . . . . . . . . 16 (𝑥 ∈ V → 𝑥 ⊆ (fi‘𝑥))
1817elv 3498 . . . . . . . . . . . . . . 15 𝑥 ⊆ (fi‘𝑥)
19 simplrl 775 . . . . . . . . . . . . . . . . . 18 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) → 𝑥 ⊆ (Clsd‘𝐽))
201cldss2 21630 . . . . . . . . . . . . . . . . . . 19 (Clsd‘𝐽) ⊆ 𝒫 𝐽
214ad2antrr 724 . . . . . . . . . . . . . . . . . . . 20 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) → 𝑋 = 𝐽)
2221pweqd 4542 . . . . . . . . . . . . . . . . . . 19 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) → 𝒫 𝑋 = 𝒫 𝐽)
2320, 22sseqtrrid 4018 . . . . . . . . . . . . . . . . . 18 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) → (Clsd‘𝐽) ⊆ 𝒫 𝑋)
2419, 23sstrd 3975 . . . . . . . . . . . . . . . . 17 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) → 𝑥 ⊆ 𝒫 𝑋)
25 simpr 487 . . . . . . . . . . . . . . . . 17 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) → 𝑥 ≠ ∅)
26 simplrr 776 . . . . . . . . . . . . . . . . 17 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) → ¬ ∅ ∈ (fi‘𝑥))
27 toponmax 21526 . . . . . . . . . . . . . . . . . . 19 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
2827ad2antrr 724 . . . . . . . . . . . . . . . . . 18 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) → 𝑋𝐽)
29 fsubbas 22467 . . . . . . . . . . . . . . . . . 18 (𝑋𝐽 → ((fi‘𝑥) ∈ (fBas‘𝑋) ↔ (𝑥 ⊆ 𝒫 𝑋𝑥 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝑥))))
3028, 29syl 17 . . . . . . . . . . . . . . . . 17 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) → ((fi‘𝑥) ∈ (fBas‘𝑋) ↔ (𝑥 ⊆ 𝒫 𝑋𝑥 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝑥))))
3124, 25, 26, 30mpbir3and 1337 . . . . . . . . . . . . . . . 16 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) → (fi‘𝑥) ∈ (fBas‘𝑋))
32 ssfg 22472 . . . . . . . . . . . . . . . 16 ((fi‘𝑥) ∈ (fBas‘𝑋) → (fi‘𝑥) ⊆ (𝑋filGen(fi‘𝑥)))
3331, 32syl 17 . . . . . . . . . . . . . . 15 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) → (fi‘𝑥) ⊆ (𝑋filGen(fi‘𝑥)))
3418, 33sstrid 3976 . . . . . . . . . . . . . 14 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) → 𝑥 ⊆ (𝑋filGen(fi‘𝑥)))
3534sselda 3965 . . . . . . . . . . . . 13 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) ∧ 𝑦𝑥) → 𝑦 ∈ (𝑋filGen(fi‘𝑥)))
36 fclssscls 22618 . . . . . . . . . . . . 13 (𝑦 ∈ (𝑋filGen(fi‘𝑥)) → (𝐽 fClus (𝑋filGen(fi‘𝑥))) ⊆ ((cls‘𝐽)‘𝑦))
3735, 36syl 17 . . . . . . . . . . . 12 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) ∧ 𝑦𝑥) → (𝐽 fClus (𝑋filGen(fi‘𝑥))) ⊆ ((cls‘𝐽)‘𝑦))
3819sselda 3965 . . . . . . . . . . . . 13 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) ∧ 𝑦𝑥) → 𝑦 ∈ (Clsd‘𝐽))
39 cldcls 21642 . . . . . . . . . . . . 13 (𝑦 ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘𝑦) = 𝑦)
4038, 39syl 17 . . . . . . . . . . . 12 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) ∧ 𝑦𝑥) → ((cls‘𝐽)‘𝑦) = 𝑦)
4137, 40sseqtrd 4005 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) ∧ 𝑦𝑥) → (𝐽 fClus (𝑋filGen(fi‘𝑥))) ⊆ 𝑦)
4241ralrimiva 3180 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) → ∀𝑦𝑥 (𝐽 fClus (𝑋filGen(fi‘𝑥))) ⊆ 𝑦)
43 ssint 4883 . . . . . . . . . 10 ((𝐽 fClus (𝑋filGen(fi‘𝑥))) ⊆ 𝑥 ↔ ∀𝑦𝑥 (𝐽 fClus (𝑋filGen(fi‘𝑥))) ⊆ 𝑦)
4442, 43sylibr 236 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) → (𝐽 fClus (𝑋filGen(fi‘𝑥))) ⊆ 𝑥)
45 fgcl 22478 . . . . . . . . . 10 ((fi‘𝑥) ∈ (fBas‘𝑋) → (𝑋filGen(fi‘𝑥)) ∈ (Fil‘𝑋))
46 oveq2 7156 . . . . . . . . . . . 12 (𝑓 = (𝑋filGen(fi‘𝑥)) → (𝐽 fClus 𝑓) = (𝐽 fClus (𝑋filGen(fi‘𝑥))))
4746neeq1d 3073 . . . . . . . . . . 11 (𝑓 = (𝑋filGen(fi‘𝑥)) → ((𝐽 fClus 𝑓) ≠ ∅ ↔ (𝐽 fClus (𝑋filGen(fi‘𝑥))) ≠ ∅))
4847rspcv 3616 . . . . . . . . . 10 ((𝑋filGen(fi‘𝑥)) ∈ (Fil‘𝑋) → (∀𝑓 ∈ (Fil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ → (𝐽 fClus (𝑋filGen(fi‘𝑥))) ≠ ∅))
4931, 45, 483syl 18 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) → (∀𝑓 ∈ (Fil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ → (𝐽 fClus (𝑋filGen(fi‘𝑥))) ≠ ∅))
50 ssn0 4352 . . . . . . . . 9 (((𝐽 fClus (𝑋filGen(fi‘𝑥))) ⊆ 𝑥 ∧ (𝐽 fClus (𝑋filGen(fi‘𝑥))) ≠ ∅) → 𝑥 ≠ ∅)
5144, 49, 50syl6an 682 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) → (∀𝑓 ∈ (Fil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ → 𝑥 ≠ ∅))
5216, 51pm2.61dane 3102 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) → (∀𝑓 ∈ (Fil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ → 𝑥 ≠ ∅))
5352expr 459 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ (Clsd‘𝐽)) → (¬ ∅ ∈ (fi‘𝑥) → (∀𝑓 ∈ (Fil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ → 𝑥 ≠ ∅)))
548, 53sylan2 594 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) → (¬ ∅ ∈ (fi‘𝑥) → (∀𝑓 ∈ (Fil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ → 𝑥 ≠ ∅)))
5554com23 86 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) → (∀𝑓 ∈ (Fil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ → (¬ ∅ ∈ (fi‘𝑥) → 𝑥 ≠ ∅)))
5655ralrimdva 3187 . . 3 (𝐽 ∈ (TopOn‘𝑋) → (∀𝑓 ∈ (Fil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ → ∀𝑥 ∈ 𝒫 (Clsd‘𝐽)(¬ ∅ ∈ (fi‘𝑥) → 𝑥 ≠ ∅)))
57 topontop 21513 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
58 cmpfi 22008 . . . 4 (𝐽 ∈ Top → (𝐽 ∈ Comp ↔ ∀𝑥 ∈ 𝒫 (Clsd‘𝐽)(¬ ∅ ∈ (fi‘𝑥) → 𝑥 ≠ ∅)))
5957, 58syl 17 . . 3 (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Comp ↔ ∀𝑥 ∈ 𝒫 (Clsd‘𝐽)(¬ ∅ ∈ (fi‘𝑥) → 𝑥 ≠ ∅)))
6056, 59sylibrd 261 . 2 (𝐽 ∈ (TopOn‘𝑋) → (∀𝑓 ∈ (Fil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ → 𝐽 ∈ Comp))
617, 60impbid 214 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 208  wa 398  w3a 1082   = wceq 1531  wcel 2108  wne 3014  wral 3136  Vcvv 3493  wss 3934  c0 4289  𝒫 cpw 4537   cuni 4830   cint 4867  cfv 6348  (class class class)co 7148  ficfi 8866  fBascfbas 20525  filGencfg 20526  Topctop 21493  TopOnctopon 21510  Clsdccld 21616  clsccl 21618  Compccmp 21986  Filcfil 22445   fClus cfcls 22536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1083  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-nel 3122  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-int 4868  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7573  df-1st 7681  df-2nd 7682  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-1o 8094  df-2o 8095  df-oadd 8098  df-er 8281  df-map 8400  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-fi 8867  df-fbas 20534  df-fg 20535  df-top 21494  df-topon 21511  df-cld 21619  df-cls 21621  df-cmp 21987  df-fil 22446  df-fcls 22541
This theorem is referenced by:  ufilcmp  22632
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