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Theorem alexsub 23943
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 23949 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 714 . . . . . . . 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 770 . . . . . . . 8 ((πœ‘ ∧ (𝑓 ∈ (UFilβ€˜π‘‹) ∧ (𝐽 fLim 𝑓) = βˆ…)) β†’ 𝑓 ∈ (UFilβ€˜π‘‹))
10 simprr 772 . . . . . . . 8 ((πœ‘ ∧ (𝑓 ∈ (UFilβ€˜π‘‹) ∧ (𝐽 fLim 𝑓) = βˆ…)) β†’ (𝐽 fLim 𝑓) = βˆ…)
112, 4, 6, 8, 9, 10alexsublem 23942 . . . . . . 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 2943 . . 3 ((πœ‘ ∧ 𝑓 ∈ (UFilβ€˜π‘‹)) β†’ (𝐽 fLim 𝑓) β‰  βˆ…)
1615ralrimiva 3142 . 2 (πœ‘ β†’ βˆ€π‘“ ∈ (UFilβ€˜π‘‹)(𝐽 fLim 𝑓) β‰  βˆ…)
17 fibas 22874 . . . . . 6 (fiβ€˜π΅) ∈ TopBases
18 tgtopon 22868 . . . . . 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 2837 . . . 4 (πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜βˆͺ (fiβ€˜π΅)))
211elexd 3491 . . . . . . . . 9 (πœ‘ β†’ 𝑋 ∈ V)
223, 21eqeltrrd 2830 . . . . . . . 8 (πœ‘ β†’ βˆͺ 𝐡 ∈ V)
23 uniexb 7761 . . . . . . . 8 (𝐡 ∈ V ↔ βˆͺ 𝐡 ∈ V)
2422, 23sylibr 233 . . . . . . 7 (πœ‘ β†’ 𝐡 ∈ V)
25 fiuni 9446 . . . . . . 7 (𝐡 ∈ V β†’ βˆͺ 𝐡 = βˆͺ (fiβ€˜π΅))
2624, 25syl 17 . . . . . 6 (πœ‘ β†’ βˆͺ 𝐡 = βˆͺ (fiβ€˜π΅))
273, 26eqtrd 2768 . . . . 5 (πœ‘ β†’ 𝑋 = βˆͺ (fiβ€˜π΅))
2827fveq2d 6896 . . . 4 (πœ‘ β†’ (TopOnβ€˜π‘‹) = (TopOnβ€˜βˆͺ (fiβ€˜π΅)))
2920, 28eleqtrrd 2832 . . 3 (πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
30 ufilcmp 23930 . . 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 1534   ∈ wcel 2099   β‰  wne 2936  βˆ€wral 3057  βˆƒwrex 3066  Vcvv 3470   ∩ cin 3944   βŠ† wss 3945  βˆ…c0 4319  π’« cpw 4599  βˆͺ cuni 4904  β€˜cfv 6543  (class class class)co 7415  Fincfn 8958  ficfi 9428  topGenctg 17413  TopOnctopon 22806  TopBasesctb 22842  Compccmp 23284  UFilcufil 23797  UFLcufl 23798   fLim cflim 23832
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5360  ax-pr 5424  ax-un 7735
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-nel 3043  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3964  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-int 4946  df-iun 4994  df-iin 4995  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5571  df-eprel 5577  df-po 5585  df-so 5586  df-fr 5628  df-we 5630  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7418  df-oprab 7419  df-mpo 7420  df-om 7866  df-1st 7988  df-2nd 7989  df-1o 8481  df-er 8719  df-en 8959  df-dom 8960  df-fin 8962  df-fi 9429  df-topgen 17419  df-fbas 21270  df-fg 21271  df-top 22790  df-topon 22807  df-bases 22843  df-cld 22917  df-ntr 22918  df-cls 22919  df-nei 22996  df-cmp 23285  df-fil 23744  df-ufil 23799  df-ufl 23800  df-flim 23837  df-fcls 23839
This theorem is referenced by:  alexsubb  23944  ptcmplem5  23954
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