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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 21665 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 480 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ (UFil‘𝑋) ∧ (𝐽 fLim 𝑓) = ∅)) → 𝑋 ∈ UFL) |
3 | alexsub.2 | . . . . . . . . 9 ⊢ (𝜑 → 𝑋 = ∪ 𝐵) | |
4 | 3 | adantr 480 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ (UFil‘𝑋) ∧ (𝐽 fLim 𝑓) = ∅)) → 𝑋 = ∪ 𝐵) |
5 | alexsub.3 | . . . . . . . . 9 ⊢ (𝜑 → 𝐽 = (topGen‘(fi‘𝐵))) | |
6 | 5 | adantr 480 | . . . . . . . 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 747 | . . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓 ∈ (UFil‘𝑋) ∧ (𝐽 fLim 𝑓) = ∅)) ∧ (𝑥 ⊆ 𝐵 ∧ 𝑋 = ∪ 𝑥)) → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = ∪ 𝑦) |
9 | simprl 790 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ (UFil‘𝑋) ∧ (𝐽 fLim 𝑓) = ∅)) → 𝑓 ∈ (UFil‘𝑋)) | |
10 | simprr 792 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ (UFil‘𝑋) ∧ (𝐽 fLim 𝑓) = ∅)) → (𝐽 fLim 𝑓) = ∅) | |
11 | 2, 4, 6, 8, 9, 10 | alexsublem 21658 | . . . . . . 7 ⊢ ¬ (𝜑 ∧ (𝑓 ∈ (UFil‘𝑋) ∧ (𝐽 fLim 𝑓) = ∅)) |
12 | 11 | pm2.21i 115 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ (UFil‘𝑋) ∧ (𝐽 fLim 𝑓) = ∅)) → ¬ (𝐽 fLim 𝑓) = ∅) |
13 | 12 | expr 641 | . . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (UFil‘𝑋)) → ((𝐽 fLim 𝑓) = ∅ → ¬ (𝐽 fLim 𝑓) = ∅)) |
14 | 13 | pm2.01d 180 | . . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (UFil‘𝑋)) → ¬ (𝐽 fLim 𝑓) = ∅) |
15 | 14 | neqned 2789 | . . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (UFil‘𝑋)) → (𝐽 fLim 𝑓) ≠ ∅) |
16 | 15 | ralrimiva 2949 | . 2 ⊢ (𝜑 → ∀𝑓 ∈ (UFil‘𝑋)(𝐽 fLim 𝑓) ≠ ∅) |
17 | fibas 20592 | . . . . . 6 ⊢ (fi‘𝐵) ∈ TopBases | |
18 | tgtopon 20586 | . . . . . 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 2696 | . . . 4 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘∪ (fi‘𝐵))) |
21 | elex 3185 | . . . . . . . . . 10 ⊢ (𝑋 ∈ UFL → 𝑋 ∈ V) | |
22 | 1, 21 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ V) |
23 | 3, 22 | eqeltrrd 2689 | . . . . . . . 8 ⊢ (𝜑 → ∪ 𝐵 ∈ V) |
24 | uniexb 6866 | . . . . . . . 8 ⊢ (𝐵 ∈ V ↔ ∪ 𝐵 ∈ V) | |
25 | 23, 24 | sylibr 223 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ V) |
26 | fiuni 8217 | . . . . . . 7 ⊢ (𝐵 ∈ V → ∪ 𝐵 = ∪ (fi‘𝐵)) | |
27 | 25, 26 | syl 17 | . . . . . 6 ⊢ (𝜑 → ∪ 𝐵 = ∪ (fi‘𝐵)) |
28 | 3, 27 | eqtrd 2644 | . . . . 5 ⊢ (𝜑 → 𝑋 = ∪ (fi‘𝐵)) |
29 | 28 | fveq2d 6107 | . . . 4 ⊢ (𝜑 → (TopOn‘𝑋) = (TopOn‘∪ (fi‘𝐵))) |
30 | 20, 29 | eleqtrrd 2691 | . . 3 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) |
31 | ufilcmp 21646 | . . 3 ⊢ ((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝐽 ∈ Comp ↔ ∀𝑓 ∈ (UFil‘𝑋)(𝐽 fLim 𝑓) ≠ ∅)) | |
32 | 1, 30, 31 | syl2anc 691 | . 2 ⊢ (𝜑 → (𝐽 ∈ Comp ↔ ∀𝑓 ∈ (UFil‘𝑋)(𝐽 fLim 𝑓) ≠ ∅)) |
33 | 16, 32 | mpbird 246 | 1 ⊢ (𝜑 → 𝐽 ∈ Comp) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 ∃wrex 2897 Vcvv 3173 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 𝒫 cpw 4108 ∪ cuni 4372 ‘cfv 5804 (class class class)co 6549 Fincfn 7841 ficfi 8199 topGenctg 15921 TopOnctopon 20518 TopBasesctb 20520 Compccmp 20999 UFilcufil 21513 UFLcufl 21514 fLim cflim 21548 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fi 8200 df-topgen 15927 df-fbas 19564 df-fg 19565 df-top 20521 df-bases 20522 df-topon 20523 df-cld 20633 df-ntr 20634 df-cls 20635 df-nei 20712 df-cmp 21000 df-fil 21460 df-ufil 21515 df-ufl 21516 df-flim 21553 df-fcls 21555 |
This theorem is referenced by: alexsubb 21660 ptcmplem5 21670 |
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