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Mirrors > Home > MPE Home > Th. List > alexsub | Structured version Visualization version GIF version |
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 22232 for a proof using Zorn's lemma. (Contributed by Jeff Hankins, 24-Jan-2010.) (Revised by Mario Carneiro, 26-Aug-2015.) |
Ref | Expression |
---|---|
alexsub.1 | ⊢ (𝜑 → 𝑋 ∈ UFL) |
alexsub.2 | ⊢ (𝜑 → 𝑋 = ∪ 𝐵) |
alexsub.3 | ⊢ (𝜑 → 𝐽 = (topGen‘(fi‘𝐵))) |
alexsub.4 | ⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐵 ∧ 𝑋 = ∪ 𝑥)) → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = ∪ 𝑦) |
Ref | Expression |
---|---|
alexsub | ⊢ (𝜑 → 𝐽 ∈ Comp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | alexsub.1 | . . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ UFL) | |
2 | 1 | adantr 474 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ (UFil‘𝑋) ∧ (𝐽 fLim 𝑓) = ∅)) → 𝑋 ∈ UFL) |
3 | alexsub.2 | . . . . . . . . 9 ⊢ (𝜑 → 𝑋 = ∪ 𝐵) | |
4 | 3 | adantr 474 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ (UFil‘𝑋) ∧ (𝐽 fLim 𝑓) = ∅)) → 𝑋 = ∪ 𝐵) |
5 | alexsub.3 | . . . . . . . . 9 ⊢ (𝜑 → 𝐽 = (topGen‘(fi‘𝐵))) | |
6 | 5 | adantr 474 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ (UFil‘𝑋) ∧ (𝐽 fLim 𝑓) = ∅)) → 𝐽 = (topGen‘(fi‘𝐵))) |
7 | alexsub.4 | . . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐵 ∧ 𝑋 = ∪ 𝑥)) → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = ∪ 𝑦) | |
8 | 7 | adantlr 706 | . . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓 ∈ (UFil‘𝑋) ∧ (𝐽 fLim 𝑓) = ∅)) ∧ (𝑥 ⊆ 𝐵 ∧ 𝑋 = ∪ 𝑥)) → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = ∪ 𝑦) |
9 | simprl 787 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ (UFil‘𝑋) ∧ (𝐽 fLim 𝑓) = ∅)) → 𝑓 ∈ (UFil‘𝑋)) | |
10 | simprr 789 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ (UFil‘𝑋) ∧ (𝐽 fLim 𝑓) = ∅)) → (𝐽 fLim 𝑓) = ∅) | |
11 | 2, 4, 6, 8, 9, 10 | alexsublem 22225 | . . . . . . 7 ⊢ ¬ (𝜑 ∧ (𝑓 ∈ (UFil‘𝑋) ∧ (𝐽 fLim 𝑓) = ∅)) |
12 | 11 | pm2.21i 117 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ (UFil‘𝑋) ∧ (𝐽 fLim 𝑓) = ∅)) → ¬ (𝐽 fLim 𝑓) = ∅) |
13 | 12 | expr 450 | . . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (UFil‘𝑋)) → ((𝐽 fLim 𝑓) = ∅ → ¬ (𝐽 fLim 𝑓) = ∅)) |
14 | 13 | pm2.01d 182 | . . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (UFil‘𝑋)) → ¬ (𝐽 fLim 𝑓) = ∅) |
15 | 14 | neqned 3006 | . . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (UFil‘𝑋)) → (𝐽 fLim 𝑓) ≠ ∅) |
16 | 15 | ralrimiva 3175 | . 2 ⊢ (𝜑 → ∀𝑓 ∈ (UFil‘𝑋)(𝐽 fLim 𝑓) ≠ ∅) |
17 | fibas 21159 | . . . . . 6 ⊢ (fi‘𝐵) ∈ TopBases | |
18 | tgtopon 21153 | . . . . . 6 ⊢ ((fi‘𝐵) ∈ TopBases → (topGen‘(fi‘𝐵)) ∈ (TopOn‘∪ (fi‘𝐵))) | |
19 | 17, 18 | ax-mp 5 | . . . . 5 ⊢ (topGen‘(fi‘𝐵)) ∈ (TopOn‘∪ (fi‘𝐵)) |
20 | 5, 19 | syl6eqel 2914 | . . . 4 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘∪ (fi‘𝐵))) |
21 | elex 3429 | . . . . . . . . . 10 ⊢ (𝑋 ∈ UFL → 𝑋 ∈ V) | |
22 | 1, 21 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ V) |
23 | 3, 22 | eqeltrrd 2907 | . . . . . . . 8 ⊢ (𝜑 → ∪ 𝐵 ∈ V) |
24 | uniexb 7238 | . . . . . . . 8 ⊢ (𝐵 ∈ V ↔ ∪ 𝐵 ∈ V) | |
25 | 23, 24 | sylibr 226 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ V) |
26 | fiuni 8609 | . . . . . . 7 ⊢ (𝐵 ∈ V → ∪ 𝐵 = ∪ (fi‘𝐵)) | |
27 | 25, 26 | syl 17 | . . . . . 6 ⊢ (𝜑 → ∪ 𝐵 = ∪ (fi‘𝐵)) |
28 | 3, 27 | eqtrd 2861 | . . . . 5 ⊢ (𝜑 → 𝑋 = ∪ (fi‘𝐵)) |
29 | 28 | fveq2d 6441 | . . . 4 ⊢ (𝜑 → (TopOn‘𝑋) = (TopOn‘∪ (fi‘𝐵))) |
30 | 20, 29 | eleqtrrd 2909 | . . 3 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) |
31 | ufilcmp 22213 | . . 3 ⊢ ((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝐽 ∈ Comp ↔ ∀𝑓 ∈ (UFil‘𝑋)(𝐽 fLim 𝑓) ≠ ∅)) | |
32 | 1, 30, 31 | syl2anc 579 | . 2 ⊢ (𝜑 → (𝐽 ∈ Comp ↔ ∀𝑓 ∈ (UFil‘𝑋)(𝐽 fLim 𝑓) ≠ ∅)) |
33 | 16, 32 | mpbird 249 | 1 ⊢ (𝜑 → 𝐽 ∈ Comp) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1656 ∈ wcel 2164 ≠ wne 2999 ∀wral 3117 ∃wrex 3118 Vcvv 3414 ∩ cin 3797 ⊆ wss 3798 ∅c0 4146 𝒫 cpw 4380 ∪ cuni 4660 ‘cfv 6127 (class class class)co 6910 Fincfn 8228 ficfi 8591 topGenctg 16458 TopOnctopon 21092 TopBasesctb 21127 Compccmp 21567 UFilcufil 22080 UFLcufl 22081 fLim cflim 22115 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-int 4700 df-iun 4744 df-iin 4745 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-om 7332 df-1st 7433 df-2nd 7434 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-1o 7831 df-2o 7832 df-oadd 7835 df-er 8014 df-map 8129 df-en 8229 df-dom 8230 df-sdom 8231 df-fin 8232 df-fi 8592 df-topgen 16464 df-fbas 20110 df-fg 20111 df-top 21076 df-topon 21093 df-bases 21128 df-cld 21201 df-ntr 21202 df-cls 21203 df-nei 21280 df-cmp 21568 df-fil 22027 df-ufil 22082 df-ufl 22083 df-flim 22120 df-fcls 22122 |
This theorem is referenced by: alexsubb 22227 ptcmplem5 22237 |
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