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Theorem alexsub 23419
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 23425 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 482 . . . . . . . 8 ((πœ‘ ∧ (𝑓 ∈ (UFilβ€˜π‘‹) ∧ (𝐽 fLim 𝑓) = βˆ…)) β†’ 𝑋 ∈ UFL)
3 alexsub.2 . . . . . . . . 9 (πœ‘ β†’ 𝑋 = βˆͺ 𝐡)
43adantr 482 . . . . . . . 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 482 . . . . . . . 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 23418 . . . . . . 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 458 . . . . 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 3140 . 2 (πœ‘ β†’ βˆ€π‘“ ∈ (UFilβ€˜π‘‹)(𝐽 fLim 𝑓) β‰  βˆ…)
17 fibas 22350 . . . . . 6 (fiβ€˜π΅) ∈ TopBases
18 tgtopon 22344 . . . . . 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 2842 . . . 4 (πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜βˆͺ (fiβ€˜π΅)))
211elexd 3467 . . . . . . . . 9 (πœ‘ β†’ 𝑋 ∈ V)
223, 21eqeltrrd 2835 . . . . . . . 8 (πœ‘ β†’ βˆͺ 𝐡 ∈ V)
23 uniexb 7702 . . . . . . . 8 (𝐡 ∈ V ↔ βˆͺ 𝐡 ∈ V)
2422, 23sylibr 233 . . . . . . 7 (πœ‘ β†’ 𝐡 ∈ V)
25 fiuni 9372 . . . . . . 7 (𝐡 ∈ V β†’ βˆͺ 𝐡 = βˆͺ (fiβ€˜π΅))
2624, 25syl 17 . . . . . 6 (πœ‘ β†’ βˆͺ 𝐡 = βˆͺ (fiβ€˜π΅))
273, 26eqtrd 2773 . . . . 5 (πœ‘ β†’ 𝑋 = βˆͺ (fiβ€˜π΅))
2827fveq2d 6850 . . . 4 (πœ‘ β†’ (TopOnβ€˜π‘‹) = (TopOnβ€˜βˆͺ (fiβ€˜π΅)))
2920, 28eleqtrrd 2837 . . 3 (πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
30 ufilcmp 23406 . . 3 ((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOnβ€˜π‘‹)) β†’ (𝐽 ∈ Comp ↔ βˆ€π‘“ ∈ (UFilβ€˜π‘‹)(𝐽 fLim 𝑓) β‰  βˆ…))
311, 29, 30syl2anc 585 . 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 397   = wceq 1542   ∈ wcel 2107   β‰  wne 2940  βˆ€wral 3061  βˆƒwrex 3070  Vcvv 3447   ∩ cin 3913   βŠ† wss 3914  βˆ…c0 4286  π’« cpw 4564  βˆͺ cuni 4869  β€˜cfv 6500  (class class class)co 7361  Fincfn 8889  ficfi 9354  topGenctg 17327  TopOnctopon 22282  TopBasesctb 22318  Compccmp 22760  UFilcufil 23273  UFLcufl 23274   fLim cflim 23308
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-int 4912  df-iun 4960  df-iin 4961  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-1st 7925  df-2nd 7926  df-1o 8416  df-er 8654  df-en 8890  df-dom 8891  df-fin 8893  df-fi 9355  df-topgen 17333  df-fbas 20816  df-fg 20817  df-top 22266  df-topon 22283  df-bases 22319  df-cld 22393  df-ntr 22394  df-cls 22395  df-nei 22472  df-cmp 22761  df-fil 23220  df-ufil 23275  df-ufl 23276  df-flim 23313  df-fcls 23315
This theorem is referenced by:  alexsubb  23420  ptcmplem5  23430
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