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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 23948 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 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 23941 | . . . . . . 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 456 | . . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (UFil‘𝑋)) → ((𝐽 fLim 𝑓) = ∅ → ¬ (𝐽 fLim 𝑓) = ∅)) |
14 | 13 | pm2.01d 189 | . . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (UFil‘𝑋)) → ¬ (𝐽 fLim 𝑓) = ∅) |
15 | 14 | neqned 2943 | . . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (UFil‘𝑋)) → (𝐽 fLim 𝑓) ≠ ∅) |
16 | 15 | ralrimiva 3142 | . 2 ⊢ (𝜑 → ∀𝑓 ∈ (UFil‘𝑋)(𝐽 fLim 𝑓) ≠ ∅) |
17 | fibas 22873 | . . . . . 6 ⊢ (fi‘𝐵) ∈ TopBases | |
18 | tgtopon 22867 | . . . . . 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 2837 | . . . 4 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘∪ (fi‘𝐵))) |
21 | 1 | elexd 3491 | . . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ V) |
22 | 3, 21 | eqeltrrd 2830 | . . . . . . . 8 ⊢ (𝜑 → ∪ 𝐵 ∈ V) |
23 | uniexb 7760 | . . . . . . . 8 ⊢ (𝐵 ∈ V ↔ ∪ 𝐵 ∈ V) | |
24 | 22, 23 | sylibr 233 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ V) |
25 | fiuni 9445 | . . . . . . 7 ⊢ (𝐵 ∈ V → ∪ 𝐵 = ∪ (fi‘𝐵)) | |
26 | 24, 25 | syl 17 | . . . . . 6 ⊢ (𝜑 → ∪ 𝐵 = ∪ (fi‘𝐵)) |
27 | 3, 26 | eqtrd 2768 | . . . . 5 ⊢ (𝜑 → 𝑋 = ∪ (fi‘𝐵)) |
28 | 27 | fveq2d 6895 | . . . 4 ⊢ (𝜑 → (TopOn‘𝑋) = (TopOn‘∪ (fi‘𝐵))) |
29 | 20, 28 | eleqtrrd 2832 | . . 3 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) |
30 | ufilcmp 23929 | . . 3 ⊢ ((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝐽 ∈ Comp ↔ ∀𝑓 ∈ (UFil‘𝑋)(𝐽 fLim 𝑓) ≠ ∅)) | |
31 | 1, 29, 30 | syl2anc 583 | . 2 ⊢ (𝜑 → (𝐽 ∈ Comp ↔ ∀𝑓 ∈ (UFil‘𝑋)(𝐽 fLim 𝑓) ≠ ∅)) |
32 | 16, 31 | mpbird 257 | 1 ⊢ (𝜑 → 𝐽 ∈ Comp) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ≠ wne 2936 ∀wral 3057 ∃wrex 3066 Vcvv 3470 ∩ cin 3944 ⊆ wss 3945 ∅c0 4318 𝒫 cpw 4598 ∪ cuni 4903 ‘cfv 6542 (class class class)co 7414 Fincfn 8957 ficfi 9427 topGenctg 17412 TopOnctopon 22805 TopBasesctb 22841 Compccmp 23283 UFilcufil 23796 UFLcufl 23797 fLim cflim 23831 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-1o 8480 df-er 8718 df-en 8958 df-dom 8959 df-fin 8961 df-fi 9428 df-topgen 17418 df-fbas 21269 df-fg 21270 df-top 22789 df-topon 22806 df-bases 22842 df-cld 22916 df-ntr 22917 df-cls 22918 df-nei 22995 df-cmp 23284 df-fil 23743 df-ufil 23798 df-ufl 23799 df-flim 23836 df-fcls 23838 |
This theorem is referenced by: alexsubb 23943 ptcmplem5 23953 |
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