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Theorem alexsub 22645
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 22651 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 483 . . . . . . . 8 ((𝜑 ∧ (𝑓 ∈ (UFil‘𝑋) ∧ (𝐽 fLim 𝑓) = ∅)) → 𝑋 ∈ UFL)
3 alexsub.2 . . . . . . . . 9 (𝜑𝑋 = 𝐵)
43adantr 483 . . . . . . . 8 ((𝜑 ∧ (𝑓 ∈ (UFil‘𝑋) ∧ (𝐽 fLim 𝑓) = ∅)) → 𝑋 = 𝐵)
5 alexsub.3 . . . . . . . . 9 (𝜑𝐽 = (topGen‘(fi‘𝐵)))
65adantr 483 . . . . . . . 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 713 . . . . . . . 8 (((𝜑 ∧ (𝑓 ∈ (UFil‘𝑋) ∧ (𝐽 fLim 𝑓) = ∅)) ∧ (𝑥𝐵𝑋 = 𝑥)) → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = 𝑦)
9 simprl 769 . . . . . . . 8 ((𝜑 ∧ (𝑓 ∈ (UFil‘𝑋) ∧ (𝐽 fLim 𝑓) = ∅)) → 𝑓 ∈ (UFil‘𝑋))
10 simprr 771 . . . . . . . 8 ((𝜑 ∧ (𝑓 ∈ (UFil‘𝑋) ∧ (𝐽 fLim 𝑓) = ∅)) → (𝐽 fLim 𝑓) = ∅)
112, 4, 6, 8, 9, 10alexsublem 22644 . . . . . . 7 ¬ (𝜑 ∧ (𝑓 ∈ (UFil‘𝑋) ∧ (𝐽 fLim 𝑓) = ∅))
1211pm2.21i 119 . . . . . 6 ((𝜑 ∧ (𝑓 ∈ (UFil‘𝑋) ∧ (𝐽 fLim 𝑓) = ∅)) → ¬ (𝐽 fLim 𝑓) = ∅)
1312expr 459 . . . . 5 ((𝜑𝑓 ∈ (UFil‘𝑋)) → ((𝐽 fLim 𝑓) = ∅ → ¬ (𝐽 fLim 𝑓) = ∅))
1413pm2.01d 192 . . . 4 ((𝜑𝑓 ∈ (UFil‘𝑋)) → ¬ (𝐽 fLim 𝑓) = ∅)
1514neqned 3021 . . 3 ((𝜑𝑓 ∈ (UFil‘𝑋)) → (𝐽 fLim 𝑓) ≠ ∅)
1615ralrimiva 3180 . 2 (𝜑 → ∀𝑓 ∈ (UFil‘𝑋)(𝐽 fLim 𝑓) ≠ ∅)
17 fibas 21577 . . . . . 6 (fi‘𝐵) ∈ TopBases
18 tgtopon 21571 . . . . . 6 ((fi‘𝐵) ∈ TopBases → (topGen‘(fi‘𝐵)) ∈ (TopOn‘ (fi‘𝐵)))
1917, 18ax-mp 5 . . . . 5 (topGen‘(fi‘𝐵)) ∈ (TopOn‘ (fi‘𝐵))
205, 19syl6eqel 2919 . . . 4 (𝜑𝐽 ∈ (TopOn‘ (fi‘𝐵)))
211elexd 3513 . . . . . . . . 9 (𝜑𝑋 ∈ V)
223, 21eqeltrrd 2912 . . . . . . . 8 (𝜑 𝐵 ∈ V)
23 uniexb 7478 . . . . . . . 8 (𝐵 ∈ V ↔ 𝐵 ∈ V)
2422, 23sylibr 236 . . . . . . 7 (𝜑𝐵 ∈ V)
25 fiuni 8884 . . . . . . 7 (𝐵 ∈ V → 𝐵 = (fi‘𝐵))
2624, 25syl 17 . . . . . 6 (𝜑 𝐵 = (fi‘𝐵))
273, 26eqtrd 2854 . . . . 5 (𝜑𝑋 = (fi‘𝐵))
2827fveq2d 6667 . . . 4 (𝜑 → (TopOn‘𝑋) = (TopOn‘ (fi‘𝐵)))
2920, 28eleqtrrd 2914 . . 3 (𝜑𝐽 ∈ (TopOn‘𝑋))
30 ufilcmp 22632 . . 3 ((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝐽 ∈ Comp ↔ ∀𝑓 ∈ (UFil‘𝑋)(𝐽 fLim 𝑓) ≠ ∅))
311, 29, 30syl2anc 586 . 2 (𝜑 → (𝐽 ∈ Comp ↔ ∀𝑓 ∈ (UFil‘𝑋)(𝐽 fLim 𝑓) ≠ ∅))
3216, 31mpbird 259 1 (𝜑𝐽 ∈ Comp)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wa 398   = wceq 1530  wcel 2107  wne 3014  wral 3136  wrex 3137  Vcvv 3493  cin 3933  wss 3934  c0 4289  𝒫 cpw 4537   cuni 4830  cfv 6348  (class class class)co 7148  Fincfn 8501  ficfi 8866  topGenctg 16703  TopOnctopon 21510  TopBasesctb 21545  Compccmp 21986  UFilcufil 22499  UFLcufl 22500   fLim cflim 22534
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  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 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  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-topgen 16709  df-fbas 20534  df-fg 20535  df-top 21494  df-topon 21511  df-bases 21546  df-cld 21619  df-ntr 21620  df-cls 21621  df-nei 21698  df-cmp 21987  df-fil 22446  df-ufil 22501  df-ufl 22502  df-flim 22539  df-fcls 22541
This theorem is referenced by:  alexsubb  22646  ptcmplem5  22656
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