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Theorem alexsub 22581
Description: The Alexander Subbase Theorem: If 𝐵 is a subbase for the topology 𝐽, and any cover taken from 𝐵 has a finite subcover, then the generated topology is compact. This proof uses the ultrafilter lemma; see alexsubALT 22587 for a proof using Zorn's lemma. (Contributed by Jeff Hankins, 24-Jan-2010.) (Revised by Mario Carneiro, 26-Aug-2015.)
Hypotheses
Ref Expression
alexsub.1 (𝜑𝑋 ∈ UFL)
alexsub.2 (𝜑𝑋 = 𝐵)
alexsub.3 (𝜑𝐽 = (topGen‘(fi‘𝐵)))
alexsub.4 ((𝜑 ∧ (𝑥𝐵𝑋 = 𝑥)) → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = 𝑦)
Assertion
Ref Expression
alexsub (𝜑𝐽 ∈ Comp)
Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐽,𝑦   𝜑,𝑥,𝑦   𝑥,𝑋,𝑦

Proof of Theorem alexsub
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 alexsub.1 . . . . . . . . 9 (𝜑𝑋 ∈ UFL)
21adantr 481 . . . . . . . 8 ((𝜑 ∧ (𝑓 ∈ (UFil‘𝑋) ∧ (𝐽 fLim 𝑓) = ∅)) → 𝑋 ∈ UFL)
3 alexsub.2 . . . . . . . . 9 (𝜑𝑋 = 𝐵)
43adantr 481 . . . . . . . 8 ((𝜑 ∧ (𝑓 ∈ (UFil‘𝑋) ∧ (𝐽 fLim 𝑓) = ∅)) → 𝑋 = 𝐵)
5 alexsub.3 . . . . . . . . 9 (𝜑𝐽 = (topGen‘(fi‘𝐵)))
65adantr 481 . . . . . . . 8 ((𝜑 ∧ (𝑓 ∈ (UFil‘𝑋) ∧ (𝐽 fLim 𝑓) = ∅)) → 𝐽 = (topGen‘(fi‘𝐵)))
7 alexsub.4 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐵𝑋 = 𝑥)) → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = 𝑦)
87adantlr 711 . . . . . . . 8 (((𝜑 ∧ (𝑓 ∈ (UFil‘𝑋) ∧ (𝐽 fLim 𝑓) = ∅)) ∧ (𝑥𝐵𝑋 = 𝑥)) → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = 𝑦)
9 simprl 767 . . . . . . . 8 ((𝜑 ∧ (𝑓 ∈ (UFil‘𝑋) ∧ (𝐽 fLim 𝑓) = ∅)) → 𝑓 ∈ (UFil‘𝑋))
10 simprr 769 . . . . . . . 8 ((𝜑 ∧ (𝑓 ∈ (UFil‘𝑋) ∧ (𝐽 fLim 𝑓) = ∅)) → (𝐽 fLim 𝑓) = ∅)
112, 4, 6, 8, 9, 10alexsublem 22580 . . . . . . 7 ¬ (𝜑 ∧ (𝑓 ∈ (UFil‘𝑋) ∧ (𝐽 fLim 𝑓) = ∅))
1211pm2.21i 119 . . . . . 6 ((𝜑 ∧ (𝑓 ∈ (UFil‘𝑋) ∧ (𝐽 fLim 𝑓) = ∅)) → ¬ (𝐽 fLim 𝑓) = ∅)
1312expr 457 . . . . 5 ((𝜑𝑓 ∈ (UFil‘𝑋)) → ((𝐽 fLim 𝑓) = ∅ → ¬ (𝐽 fLim 𝑓) = ∅))
1413pm2.01d 191 . . . 4 ((𝜑𝑓 ∈ (UFil‘𝑋)) → ¬ (𝐽 fLim 𝑓) = ∅)
1514neqned 3020 . . 3 ((𝜑𝑓 ∈ (UFil‘𝑋)) → (𝐽 fLim 𝑓) ≠ ∅)
1615ralrimiva 3179 . 2 (𝜑 → ∀𝑓 ∈ (UFil‘𝑋)(𝐽 fLim 𝑓) ≠ ∅)
17 fibas 21513 . . . . . 6 (fi‘𝐵) ∈ TopBases
18 tgtopon 21507 . . . . . 6 ((fi‘𝐵) ∈ TopBases → (topGen‘(fi‘𝐵)) ∈ (TopOn‘ (fi‘𝐵)))
1917, 18ax-mp 5 . . . . 5 (topGen‘(fi‘𝐵)) ∈ (TopOn‘ (fi‘𝐵))
205, 19syl6eqel 2918 . . . 4 (𝜑𝐽 ∈ (TopOn‘ (fi‘𝐵)))
211elexd 3512 . . . . . . . . 9 (𝜑𝑋 ∈ V)
223, 21eqeltrrd 2911 . . . . . . . 8 (𝜑 𝐵 ∈ V)
23 uniexb 7475 . . . . . . . 8 (𝐵 ∈ V ↔ 𝐵 ∈ V)
2422, 23sylibr 235 . . . . . . 7 (𝜑𝐵 ∈ V)
25 fiuni 8880 . . . . . . 7 (𝐵 ∈ V → 𝐵 = (fi‘𝐵))
2624, 25syl 17 . . . . . 6 (𝜑 𝐵 = (fi‘𝐵))
273, 26eqtrd 2853 . . . . 5 (𝜑𝑋 = (fi‘𝐵))
2827fveq2d 6667 . . . 4 (𝜑 → (TopOn‘𝑋) = (TopOn‘ (fi‘𝐵)))
2920, 28eleqtrrd 2913 . . 3 (𝜑𝐽 ∈ (TopOn‘𝑋))
30 ufilcmp 22568 . . 3 ((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝐽 ∈ Comp ↔ ∀𝑓 ∈ (UFil‘𝑋)(𝐽 fLim 𝑓) ≠ ∅))
311, 29, 30syl2anc 584 . 2 (𝜑 → (𝐽 ∈ Comp ↔ ∀𝑓 ∈ (UFil‘𝑋)(𝐽 fLim 𝑓) ≠ ∅))
3216, 31mpbird 258 1 (𝜑𝐽 ∈ Comp)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wne 3013  wral 3135  wrex 3136  Vcvv 3492  cin 3932  wss 3933  c0 4288  𝒫 cpw 4535   cuni 4830  cfv 6348  (class class class)co 7145  Fincfn 8497  ficfi 8862  topGenctg 16699  TopOnctopon 21446  TopBasesctb 21481  Compccmp 21922  UFilcufil 22435  UFLcufl 22436   fLim cflim 22470
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  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 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-2o 8092  df-oadd 8095  df-er 8278  df-map 8397  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-fi 8863  df-topgen 16705  df-fbas 20470  df-fg 20471  df-top 21430  df-topon 21447  df-bases 21482  df-cld 21555  df-ntr 21556  df-cls 21557  df-nei 21634  df-cmp 21923  df-fil 22382  df-ufil 22437  df-ufl 22438  df-flim 22475  df-fcls 22477
This theorem is referenced by:  alexsubb  22582  ptcmplem5  22592
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