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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 22656 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 484 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ (UFil‘𝑋) ∧ (𝐽 fLim 𝑓) = ∅)) → 𝑋 ∈ UFL) |
3 | alexsub.2 | . . . . . . . . 9 ⊢ (𝜑 → 𝑋 = ∪ 𝐵) | |
4 | 3 | adantr 484 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ (UFil‘𝑋) ∧ (𝐽 fLim 𝑓) = ∅)) → 𝑋 = ∪ 𝐵) |
5 | alexsub.3 | . . . . . . . . 9 ⊢ (𝜑 → 𝐽 = (topGen‘(fi‘𝐵))) | |
6 | 5 | adantr 484 | . . . . . . . 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 714 | . . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓 ∈ (UFil‘𝑋) ∧ (𝐽 fLim 𝑓) = ∅)) ∧ (𝑥 ⊆ 𝐵 ∧ 𝑋 = ∪ 𝑥)) → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = ∪ 𝑦) |
9 | simprl 770 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ (UFil‘𝑋) ∧ (𝐽 fLim 𝑓) = ∅)) → 𝑓 ∈ (UFil‘𝑋)) | |
10 | simprr 772 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ (UFil‘𝑋) ∧ (𝐽 fLim 𝑓) = ∅)) → (𝐽 fLim 𝑓) = ∅) | |
11 | 2, 4, 6, 8, 9, 10 | alexsublem 22649 | . . . . . . 7 ⊢ ¬ (𝜑 ∧ (𝑓 ∈ (UFil‘𝑋) ∧ (𝐽 fLim 𝑓) = ∅)) |
12 | 11 | pm2.21i 119 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ (UFil‘𝑋) ∧ (𝐽 fLim 𝑓) = ∅)) → ¬ (𝐽 fLim 𝑓) = ∅) |
13 | 12 | expr 460 | . . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (UFil‘𝑋)) → ((𝐽 fLim 𝑓) = ∅ → ¬ (𝐽 fLim 𝑓) = ∅)) |
14 | 13 | pm2.01d 193 | . . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (UFil‘𝑋)) → ¬ (𝐽 fLim 𝑓) = ∅) |
15 | 14 | neqned 2994 | . . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (UFil‘𝑋)) → (𝐽 fLim 𝑓) ≠ ∅) |
16 | 15 | ralrimiva 3149 | . 2 ⊢ (𝜑 → ∀𝑓 ∈ (UFil‘𝑋)(𝐽 fLim 𝑓) ≠ ∅) |
17 | fibas 21582 | . . . . . 6 ⊢ (fi‘𝐵) ∈ TopBases | |
18 | tgtopon 21576 | . . . . . 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 | eqeltrdi 2898 | . . . 4 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘∪ (fi‘𝐵))) |
21 | 1 | elexd 3461 | . . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ V) |
22 | 3, 21 | eqeltrrd 2891 | . . . . . . . 8 ⊢ (𝜑 → ∪ 𝐵 ∈ V) |
23 | uniexb 7466 | . . . . . . . 8 ⊢ (𝐵 ∈ V ↔ ∪ 𝐵 ∈ V) | |
24 | 22, 23 | sylibr 237 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ V) |
25 | fiuni 8876 | . . . . . . 7 ⊢ (𝐵 ∈ V → ∪ 𝐵 = ∪ (fi‘𝐵)) | |
26 | 24, 25 | syl 17 | . . . . . 6 ⊢ (𝜑 → ∪ 𝐵 = ∪ (fi‘𝐵)) |
27 | 3, 26 | eqtrd 2833 | . . . . 5 ⊢ (𝜑 → 𝑋 = ∪ (fi‘𝐵)) |
28 | 27 | fveq2d 6649 | . . . 4 ⊢ (𝜑 → (TopOn‘𝑋) = (TopOn‘∪ (fi‘𝐵))) |
29 | 20, 28 | eleqtrrd 2893 | . . 3 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) |
30 | ufilcmp 22637 | . . 3 ⊢ ((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝐽 ∈ Comp ↔ ∀𝑓 ∈ (UFil‘𝑋)(𝐽 fLim 𝑓) ≠ ∅)) | |
31 | 1, 29, 30 | syl2anc 587 | . 2 ⊢ (𝜑 → (𝐽 ∈ Comp ↔ ∀𝑓 ∈ (UFil‘𝑋)(𝐽 fLim 𝑓) ≠ ∅)) |
32 | 16, 31 | mpbird 260 | 1 ⊢ (𝜑 → 𝐽 ∈ Comp) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∀wral 3106 ∃wrex 3107 Vcvv 3441 ∩ cin 3880 ⊆ wss 3881 ∅c0 4243 𝒫 cpw 4497 ∪ cuni 4800 ‘cfv 6324 (class class class)co 7135 Fincfn 8492 ficfi 8858 topGenctg 16703 TopOnctopon 21515 TopBasesctb 21550 Compccmp 21991 UFilcufil 22504 UFLcufl 22505 fLim cflim 22539 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fi 8859 df-topgen 16709 df-fbas 20088 df-fg 20089 df-top 21499 df-topon 21516 df-bases 21551 df-cld 21624 df-ntr 21625 df-cls 21626 df-nei 21703 df-cmp 21992 df-fil 22451 df-ufil 22506 df-ufl 22507 df-flim 22544 df-fcls 22546 |
This theorem is referenced by: alexsubb 22651 ptcmplem5 22661 |
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