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Theorem alexsub 23548
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 23554 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 481 . . . . . . . 8 ((πœ‘ ∧ (𝑓 ∈ (UFilβ€˜π‘‹) ∧ (𝐽 fLim 𝑓) = βˆ…)) β†’ 𝑋 ∈ UFL)
3 alexsub.2 . . . . . . . . 9 (πœ‘ β†’ 𝑋 = βˆͺ 𝐡)
43adantr 481 . . . . . . . 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 481 . . . . . . . 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 713 . . . . . . . 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 769 . . . . . . . 8 ((πœ‘ ∧ (𝑓 ∈ (UFilβ€˜π‘‹) ∧ (𝐽 fLim 𝑓) = βˆ…)) β†’ 𝑓 ∈ (UFilβ€˜π‘‹))
10 simprr 771 . . . . . . . 8 ((πœ‘ ∧ (𝑓 ∈ (UFilβ€˜π‘‹) ∧ (𝐽 fLim 𝑓) = βˆ…)) β†’ (𝐽 fLim 𝑓) = βˆ…)
112, 4, 6, 8, 9, 10alexsublem 23547 . . . . . . 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 457 . . . . 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 2947 . . 3 ((πœ‘ ∧ 𝑓 ∈ (UFilβ€˜π‘‹)) β†’ (𝐽 fLim 𝑓) β‰  βˆ…)
1615ralrimiva 3146 . 2 (πœ‘ β†’ βˆ€π‘“ ∈ (UFilβ€˜π‘‹)(𝐽 fLim 𝑓) β‰  βˆ…)
17 fibas 22479 . . . . . 6 (fiβ€˜π΅) ∈ TopBases
18 tgtopon 22473 . . . . . 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 2841 . . . 4 (πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜βˆͺ (fiβ€˜π΅)))
211elexd 3494 . . . . . . . . 9 (πœ‘ β†’ 𝑋 ∈ V)
223, 21eqeltrrd 2834 . . . . . . . 8 (πœ‘ β†’ βˆͺ 𝐡 ∈ V)
23 uniexb 7750 . . . . . . . 8 (𝐡 ∈ V ↔ βˆͺ 𝐡 ∈ V)
2422, 23sylibr 233 . . . . . . 7 (πœ‘ β†’ 𝐡 ∈ V)
25 fiuni 9422 . . . . . . 7 (𝐡 ∈ V β†’ βˆͺ 𝐡 = βˆͺ (fiβ€˜π΅))
2624, 25syl 17 . . . . . 6 (πœ‘ β†’ βˆͺ 𝐡 = βˆͺ (fiβ€˜π΅))
273, 26eqtrd 2772 . . . . 5 (πœ‘ β†’ 𝑋 = βˆͺ (fiβ€˜π΅))
2827fveq2d 6895 . . . 4 (πœ‘ β†’ (TopOnβ€˜π‘‹) = (TopOnβ€˜βˆͺ (fiβ€˜π΅)))
2920, 28eleqtrrd 2836 . . 3 (πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
30 ufilcmp 23535 . . 3 ((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOnβ€˜π‘‹)) β†’ (𝐽 ∈ Comp ↔ βˆ€π‘“ ∈ (UFilβ€˜π‘‹)(𝐽 fLim 𝑓) β‰  βˆ…))
311, 29, 30syl2anc 584 . 2 (πœ‘ β†’ (𝐽 ∈ Comp ↔ βˆ€π‘“ ∈ (UFilβ€˜π‘‹)(𝐽 fLim 𝑓) β‰  βˆ…))
3216, 31mpbird 256 1 (πœ‘ β†’ 𝐽 ∈ Comp)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106   β‰  wne 2940  βˆ€wral 3061  βˆƒwrex 3070  Vcvv 3474   ∩ cin 3947   βŠ† wss 3948  βˆ…c0 4322  π’« cpw 4602  βˆͺ cuni 4908  β€˜cfv 6543  (class class class)co 7408  Fincfn 8938  ficfi 9404  topGenctg 17382  TopOnctopon 22411  TopBasesctb 22447  Compccmp 22889  UFilcufil 23402  UFLcufl 23403   fLim cflim 23437
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  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 7411  df-oprab 7412  df-mpo 7413  df-om 7855  df-1st 7974  df-2nd 7975  df-1o 8465  df-er 8702  df-en 8939  df-dom 8940  df-fin 8942  df-fi 9405  df-topgen 17388  df-fbas 20940  df-fg 20941  df-top 22395  df-topon 22412  df-bases 22448  df-cld 22522  df-ntr 22523  df-cls 22524  df-nei 22601  df-cmp 22890  df-fil 23349  df-ufil 23404  df-ufl 23405  df-flim 23442  df-fcls 23444
This theorem is referenced by:  alexsubb  23549  ptcmplem5  23559
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