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Theorem alexsub 23873
Description: The Alexander Subbase Theorem: If 𝐡 is a subbase for the topology 𝐽, and any cover taken from 𝐡 has a finite subcover, then the generated topology is compact. This proof uses the ultrafilter lemma; see alexsubALT 23879 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 (πœ‘ β†’ 𝑋 ∈ UFL)
alexsub.2 (πœ‘ β†’ 𝑋 = βˆͺ 𝐡)
alexsub.3 (πœ‘ β†’ 𝐽 = (topGenβ€˜(fiβ€˜π΅)))
alexsub.4 ((πœ‘ ∧ (π‘₯ βŠ† 𝐡 ∧ 𝑋 = βˆͺ π‘₯)) β†’ βˆƒπ‘¦ ∈ (𝒫 π‘₯ ∩ Fin)𝑋 = βˆͺ 𝑦)
Assertion
Ref Expression
alexsub (πœ‘ β†’ 𝐽 ∈ Comp)
Distinct variable groups:   π‘₯,𝑦,𝐡   π‘₯,𝐽,𝑦   πœ‘,π‘₯,𝑦   π‘₯,𝑋,𝑦

Proof of Theorem alexsub
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 alexsub.1 . . . . . . . . 9 (πœ‘ β†’ 𝑋 ∈ UFL)
21adantr 480 . . . . . . . 8 ((πœ‘ ∧ (𝑓 ∈ (UFilβ€˜π‘‹) ∧ (𝐽 fLim 𝑓) = βˆ…)) β†’ 𝑋 ∈ UFL)
3 alexsub.2 . . . . . . . . 9 (πœ‘ β†’ 𝑋 = βˆͺ 𝐡)
43adantr 480 . . . . . . . 8 ((πœ‘ ∧ (𝑓 ∈ (UFilβ€˜π‘‹) ∧ (𝐽 fLim 𝑓) = βˆ…)) β†’ 𝑋 = βˆͺ 𝐡)
5 alexsub.3 . . . . . . . . 9 (πœ‘ β†’ 𝐽 = (topGenβ€˜(fiβ€˜π΅)))
65adantr 480 . . . . . . . 8 ((πœ‘ ∧ (𝑓 ∈ (UFilβ€˜π‘‹) ∧ (𝐽 fLim 𝑓) = βˆ…)) β†’ 𝐽 = (topGenβ€˜(fiβ€˜π΅)))
7 alexsub.4 . . . . . . . . 9 ((πœ‘ ∧ (π‘₯ βŠ† 𝐡 ∧ 𝑋 = βˆͺ π‘₯)) β†’ βˆƒπ‘¦ ∈ (𝒫 π‘₯ ∩ Fin)𝑋 = βˆͺ 𝑦)
87adantlr 712 . . . . . . . 8 (((πœ‘ ∧ (𝑓 ∈ (UFilβ€˜π‘‹) ∧ (𝐽 fLim 𝑓) = βˆ…)) ∧ (π‘₯ βŠ† 𝐡 ∧ 𝑋 = βˆͺ π‘₯)) β†’ βˆƒπ‘¦ ∈ (𝒫 π‘₯ ∩ Fin)𝑋 = βˆͺ 𝑦)
9 simprl 768 . . . . . . . 8 ((πœ‘ ∧ (𝑓 ∈ (UFilβ€˜π‘‹) ∧ (𝐽 fLim 𝑓) = βˆ…)) β†’ 𝑓 ∈ (UFilβ€˜π‘‹))
10 simprr 770 . . . . . . . 8 ((πœ‘ ∧ (𝑓 ∈ (UFilβ€˜π‘‹) ∧ (𝐽 fLim 𝑓) = βˆ…)) β†’ (𝐽 fLim 𝑓) = βˆ…)
112, 4, 6, 8, 9, 10alexsublem 23872 . . . . . . 7 Β¬ (πœ‘ ∧ (𝑓 ∈ (UFilβ€˜π‘‹) ∧ (𝐽 fLim 𝑓) = βˆ…))
1211pm2.21i 119 . . . . . 6 ((πœ‘ ∧ (𝑓 ∈ (UFilβ€˜π‘‹) ∧ (𝐽 fLim 𝑓) = βˆ…)) β†’ Β¬ (𝐽 fLim 𝑓) = βˆ…)
1312expr 456 . . . . 5 ((πœ‘ ∧ 𝑓 ∈ (UFilβ€˜π‘‹)) β†’ ((𝐽 fLim 𝑓) = βˆ… β†’ Β¬ (𝐽 fLim 𝑓) = βˆ…))
1413pm2.01d 189 . . . 4 ((πœ‘ ∧ 𝑓 ∈ (UFilβ€˜π‘‹)) β†’ Β¬ (𝐽 fLim 𝑓) = βˆ…)
1514neqned 2939 . . 3 ((πœ‘ ∧ 𝑓 ∈ (UFilβ€˜π‘‹)) β†’ (𝐽 fLim 𝑓) β‰  βˆ…)
1615ralrimiva 3138 . 2 (πœ‘ β†’ βˆ€π‘“ ∈ (UFilβ€˜π‘‹)(𝐽 fLim 𝑓) β‰  βˆ…)
17 fibas 22804 . . . . . 6 (fiβ€˜π΅) ∈ TopBases
18 tgtopon 22798 . . . . . 6 ((fiβ€˜π΅) ∈ TopBases β†’ (topGenβ€˜(fiβ€˜π΅)) ∈ (TopOnβ€˜βˆͺ (fiβ€˜π΅)))
1917, 18ax-mp 5 . . . . 5 (topGenβ€˜(fiβ€˜π΅)) ∈ (TopOnβ€˜βˆͺ (fiβ€˜π΅))
205, 19eqeltrdi 2833 . . . 4 (πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜βˆͺ (fiβ€˜π΅)))
211elexd 3487 . . . . . . . . 9 (πœ‘ β†’ 𝑋 ∈ V)
223, 21eqeltrrd 2826 . . . . . . . 8 (πœ‘ β†’ βˆͺ 𝐡 ∈ V)
23 uniexb 7745 . . . . . . . 8 (𝐡 ∈ V ↔ βˆͺ 𝐡 ∈ V)
2422, 23sylibr 233 . . . . . . 7 (πœ‘ β†’ 𝐡 ∈ V)
25 fiuni 9420 . . . . . . 7 (𝐡 ∈ V β†’ βˆͺ 𝐡 = βˆͺ (fiβ€˜π΅))
2624, 25syl 17 . . . . . 6 (πœ‘ β†’ βˆͺ 𝐡 = βˆͺ (fiβ€˜π΅))
273, 26eqtrd 2764 . . . . 5 (πœ‘ β†’ 𝑋 = βˆͺ (fiβ€˜π΅))
2827fveq2d 6886 . . . 4 (πœ‘ β†’ (TopOnβ€˜π‘‹) = (TopOnβ€˜βˆͺ (fiβ€˜π΅)))
2920, 28eleqtrrd 2828 . . 3 (πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
30 ufilcmp 23860 . . 3 ((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOnβ€˜π‘‹)) β†’ (𝐽 ∈ Comp ↔ βˆ€π‘“ ∈ (UFilβ€˜π‘‹)(𝐽 fLim 𝑓) β‰  βˆ…))
311, 29, 30syl2anc 583 . 2 (πœ‘ β†’ (𝐽 ∈ Comp ↔ βˆ€π‘“ ∈ (UFilβ€˜π‘‹)(𝐽 fLim 𝑓) β‰  βˆ…))
3216, 31mpbird 257 1 (πœ‘ β†’ 𝐽 ∈ Comp)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098   β‰  wne 2932  βˆ€wral 3053  βˆƒwrex 3062  Vcvv 3466   ∩ cin 3940   βŠ† wss 3941  βˆ…c0 4315  π’« cpw 4595  βˆͺ cuni 4900  β€˜cfv 6534  (class class class)co 7402  Fincfn 8936  ficfi 9402  topGenctg 17384  TopOnctopon 22736  TopBasesctb 22772  Compccmp 23214  UFilcufil 23727  UFLcufl 23728   fLim cflim 23762
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3960  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-int 4942  df-iun 4990  df-iin 4991  df-br 5140  df-opab 5202  df-mpt 5223  df-tr 5257  df-id 5565  df-eprel 5571  df-po 5579  df-so 5580  df-fr 5622  df-we 5624  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-ov 7405  df-oprab 7406  df-mpo 7407  df-om 7850  df-1st 7969  df-2nd 7970  df-1o 8462  df-er 8700  df-en 8937  df-dom 8938  df-fin 8940  df-fi 9403  df-topgen 17390  df-fbas 21227  df-fg 21228  df-top 22720  df-topon 22737  df-bases 22773  df-cld 22847  df-ntr 22848  df-cls 22849  df-nei 22926  df-cmp 23215  df-fil 23674  df-ufil 23729  df-ufl 23730  df-flim 23767  df-fcls 23769
This theorem is referenced by:  alexsubb  23874  ptcmplem5  23884
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