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Theorem alexsub 24069
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 24075 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 480 . . . . . . . 8 ((𝜑 ∧ (𝑓 ∈ (UFil‘𝑋) ∧ (𝐽 fLim 𝑓) = ∅)) → 𝑋 ∈ UFL)
3 alexsub.2 . . . . . . . . 9 (𝜑𝑋 = 𝐵)
43adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑓 ∈ (UFil‘𝑋) ∧ (𝐽 fLim 𝑓) = ∅)) → 𝑋 = 𝐵)
5 alexsub.3 . . . . . . . . 9 (𝜑𝐽 = (topGen‘(fi‘𝐵)))
65adantr 480 . . . . . . . 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 715 . . . . . . . 8 (((𝜑 ∧ (𝑓 ∈ (UFil‘𝑋) ∧ (𝐽 fLim 𝑓) = ∅)) ∧ (𝑥𝐵𝑋 = 𝑥)) → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = 𝑦)
9 simprl 771 . . . . . . . 8 ((𝜑 ∧ (𝑓 ∈ (UFil‘𝑋) ∧ (𝐽 fLim 𝑓) = ∅)) → 𝑓 ∈ (UFil‘𝑋))
10 simprr 773 . . . . . . . 8 ((𝜑 ∧ (𝑓 ∈ (UFil‘𝑋) ∧ (𝐽 fLim 𝑓) = ∅)) → (𝐽 fLim 𝑓) = ∅)
112, 4, 6, 8, 9, 10alexsublem 24068 . . . . . . 7 ¬ (𝜑 ∧ (𝑓 ∈ (UFil‘𝑋) ∧ (𝐽 fLim 𝑓) = ∅))
1211pm2.21i 119 . . . . . 6 ((𝜑 ∧ (𝑓 ∈ (UFil‘𝑋) ∧ (𝐽 fLim 𝑓) = ∅)) → ¬ (𝐽 fLim 𝑓) = ∅)
1312expr 456 . . . . 5 ((𝜑𝑓 ∈ (UFil‘𝑋)) → ((𝐽 fLim 𝑓) = ∅ → ¬ (𝐽 fLim 𝑓) = ∅))
1413pm2.01d 190 . . . 4 ((𝜑𝑓 ∈ (UFil‘𝑋)) → ¬ (𝐽 fLim 𝑓) = ∅)
1514neqned 2945 . . 3 ((𝜑𝑓 ∈ (UFil‘𝑋)) → (𝐽 fLim 𝑓) ≠ ∅)
1615ralrimiva 3144 . 2 (𝜑 → ∀𝑓 ∈ (UFil‘𝑋)(𝐽 fLim 𝑓) ≠ ∅)
17 fibas 23000 . . . . . 6 (fi‘𝐵) ∈ TopBases
18 tgtopon 22994 . . . . . 6 ((fi‘𝐵) ∈ TopBases → (topGen‘(fi‘𝐵)) ∈ (TopOn‘ (fi‘𝐵)))
1917, 18ax-mp 5 . . . . 5 (topGen‘(fi‘𝐵)) ∈ (TopOn‘ (fi‘𝐵))
205, 19eqeltrdi 2847 . . . 4 (𝜑𝐽 ∈ (TopOn‘ (fi‘𝐵)))
211elexd 3502 . . . . . . . . 9 (𝜑𝑋 ∈ V)
223, 21eqeltrrd 2840 . . . . . . . 8 (𝜑 𝐵 ∈ V)
23 uniexb 7783 . . . . . . . 8 (𝐵 ∈ V ↔ 𝐵 ∈ V)
2422, 23sylibr 234 . . . . . . 7 (𝜑𝐵 ∈ V)
25 fiuni 9466 . . . . . . 7 (𝐵 ∈ V → 𝐵 = (fi‘𝐵))
2624, 25syl 17 . . . . . 6 (𝜑 𝐵 = (fi‘𝐵))
273, 26eqtrd 2775 . . . . 5 (𝜑𝑋 = (fi‘𝐵))
2827fveq2d 6911 . . . 4 (𝜑 → (TopOn‘𝑋) = (TopOn‘ (fi‘𝐵)))
2920, 28eleqtrrd 2842 . . 3 (𝜑𝐽 ∈ (TopOn‘𝑋))
30 ufilcmp 24056 . . 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 257 1 (𝜑𝐽 ∈ Comp)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wne 2938  wral 3059  wrex 3068  Vcvv 3478  cin 3962  wss 3963  c0 4339  𝒫 cpw 4605   cuni 4912  cfv 6563  (class class class)co 7431  Fincfn 8984  ficfi 9448  topGenctg 17484  TopOnctopon 22932  TopBasesctb 22968  Compccmp 23410  UFilcufil 23923  UFLcufl 23924   fLim cflim 23958
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-1o 8505  df-2o 8506  df-en 8985  df-dom 8986  df-fin 8988  df-fi 9449  df-topgen 17490  df-fbas 21379  df-fg 21380  df-top 22916  df-topon 22933  df-bases 22969  df-cld 23043  df-ntr 23044  df-cls 23045  df-nei 23122  df-cmp 23411  df-fil 23870  df-ufil 23925  df-ufl 23926  df-flim 23963  df-fcls 23965
This theorem is referenced by:  alexsubb  24070  ptcmplem5  24080
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