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Theorem alexsub 22069
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 22075 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 466 . . . . . . . 8 ((𝜑 ∧ (𝑓 ∈ (UFil‘𝑋) ∧ (𝐽 fLim 𝑓) = ∅)) → 𝑋 ∈ UFL)
3 alexsub.2 . . . . . . . . 9 (𝜑𝑋 = 𝐵)
43adantr 466 . . . . . . . 8 ((𝜑 ∧ (𝑓 ∈ (UFil‘𝑋) ∧ (𝐽 fLim 𝑓) = ∅)) → 𝑋 = 𝐵)
5 alexsub.3 . . . . . . . . 9 (𝜑𝐽 = (topGen‘(fi‘𝐵)))
65adantr 466 . . . . . . . 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 694 . . . . . . . 8 (((𝜑 ∧ (𝑓 ∈ (UFil‘𝑋) ∧ (𝐽 fLim 𝑓) = ∅)) ∧ (𝑥𝐵𝑋 = 𝑥)) → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = 𝑦)
9 simprl 754 . . . . . . . 8 ((𝜑 ∧ (𝑓 ∈ (UFil‘𝑋) ∧ (𝐽 fLim 𝑓) = ∅)) → 𝑓 ∈ (UFil‘𝑋))
10 simprr 756 . . . . . . . 8 ((𝜑 ∧ (𝑓 ∈ (UFil‘𝑋) ∧ (𝐽 fLim 𝑓) = ∅)) → (𝐽 fLim 𝑓) = ∅)
112, 4, 6, 8, 9, 10alexsublem 22068 . . . . . . 7 ¬ (𝜑 ∧ (𝑓 ∈ (UFil‘𝑋) ∧ (𝐽 fLim 𝑓) = ∅))
1211pm2.21i 117 . . . . . 6 ((𝜑 ∧ (𝑓 ∈ (UFil‘𝑋) ∧ (𝐽 fLim 𝑓) = ∅)) → ¬ (𝐽 fLim 𝑓) = ∅)
1312expr 444 . . . . 5 ((𝜑𝑓 ∈ (UFil‘𝑋)) → ((𝐽 fLim 𝑓) = ∅ → ¬ (𝐽 fLim 𝑓) = ∅))
1413pm2.01d 181 . . . 4 ((𝜑𝑓 ∈ (UFil‘𝑋)) → ¬ (𝐽 fLim 𝑓) = ∅)
1514neqned 2950 . . 3 ((𝜑𝑓 ∈ (UFil‘𝑋)) → (𝐽 fLim 𝑓) ≠ ∅)
1615ralrimiva 3115 . 2 (𝜑 → ∀𝑓 ∈ (UFil‘𝑋)(𝐽 fLim 𝑓) ≠ ∅)
17 fibas 21002 . . . . . 6 (fi‘𝐵) ∈ TopBases
18 tgtopon 20996 . . . . . 6 ((fi‘𝐵) ∈ TopBases → (topGen‘(fi‘𝐵)) ∈ (TopOn‘ (fi‘𝐵)))
1917, 18ax-mp 5 . . . . 5 (topGen‘(fi‘𝐵)) ∈ (TopOn‘ (fi‘𝐵))
205, 19syl6eqel 2858 . . . 4 (𝜑𝐽 ∈ (TopOn‘ (fi‘𝐵)))
21 elex 3364 . . . . . . . . . 10 (𝑋 ∈ UFL → 𝑋 ∈ V)
221, 21syl 17 . . . . . . . . 9 (𝜑𝑋 ∈ V)
233, 22eqeltrrd 2851 . . . . . . . 8 (𝜑 𝐵 ∈ V)
24 uniexb 7120 . . . . . . . 8 (𝐵 ∈ V ↔ 𝐵 ∈ V)
2523, 24sylibr 224 . . . . . . 7 (𝜑𝐵 ∈ V)
26 fiuni 8490 . . . . . . 7 (𝐵 ∈ V → 𝐵 = (fi‘𝐵))
2725, 26syl 17 . . . . . 6 (𝜑 𝐵 = (fi‘𝐵))
283, 27eqtrd 2805 . . . . 5 (𝜑𝑋 = (fi‘𝐵))
2928fveq2d 6336 . . . 4 (𝜑 → (TopOn‘𝑋) = (TopOn‘ (fi‘𝐵)))
3020, 29eleqtrrd 2853 . . 3 (𝜑𝐽 ∈ (TopOn‘𝑋))
31 ufilcmp 22056 . . 3 ((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝐽 ∈ Comp ↔ ∀𝑓 ∈ (UFil‘𝑋)(𝐽 fLim 𝑓) ≠ ∅))
321, 30, 31syl2anc 573 . 2 (𝜑 → (𝐽 ∈ Comp ↔ ∀𝑓 ∈ (UFil‘𝑋)(𝐽 fLim 𝑓) ≠ ∅))
3316, 32mpbird 247 1 (𝜑𝐽 ∈ Comp)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  wne 2943  wral 3061  wrex 3062  Vcvv 3351  cin 3722  wss 3723  c0 4063  𝒫 cpw 4297   cuni 4574  cfv 6031  (class class class)co 6793  Fincfn 8109  ficfi 8472  topGenctg 16306  TopOnctopon 20935  TopBasesctb 20970  Compccmp 21410  UFilcufil 21923  UFLcufl 21924   fLim cflim 21958
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 837  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-topgen 16312  df-fbas 19958  df-fg 19959  df-top 20919  df-topon 20936  df-bases 20971  df-cld 21044  df-ntr 21045  df-cls 21046  df-nei 21123  df-cmp 21411  df-fil 21870  df-ufil 21925  df-ufl 21926  df-flim 21963  df-fcls 21965
This theorem is referenced by:  alexsubb  22070  ptcmplem5  22080
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