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Mirrors > Home > MPE Home > Th. List > kqt0 | Structured version Visualization version GIF version |
Description: The Kolmogorov quotient is T0 even if the original topology is not. (Contributed by Mario Carneiro, 25-Aug-2015.) |
Ref | Expression |
---|---|
kqt0 | ⊢ (𝐽 ∈ Top ↔ (KQ‘𝐽) ∈ Kol2) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | toptopon2 21454 | . . 3 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽)) | |
2 | eqid 2818 | . . . 4 ⊢ (𝑥 ∈ ∪ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}) = (𝑥 ∈ ∪ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}) | |
3 | 2 | kqt0lem 22272 | . . 3 ⊢ (𝐽 ∈ (TopOn‘∪ 𝐽) → (KQ‘𝐽) ∈ Kol2) |
4 | 1, 3 | sylbi 218 | . 2 ⊢ (𝐽 ∈ Top → (KQ‘𝐽) ∈ Kol2) |
5 | t0top 21865 | . . 3 ⊢ ((KQ‘𝐽) ∈ Kol2 → (KQ‘𝐽) ∈ Top) | |
6 | kqtop 22281 | . . 3 ⊢ (𝐽 ∈ Top ↔ (KQ‘𝐽) ∈ Top) | |
7 | 5, 6 | sylibr 235 | . 2 ⊢ ((KQ‘𝐽) ∈ Kol2 → 𝐽 ∈ Top) |
8 | 4, 7 | impbii 210 | 1 ⊢ (𝐽 ∈ Top ↔ (KQ‘𝐽) ∈ Kol2) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∈ wcel 2105 {crab 3139 ∪ cuni 4830 ↦ cmpt 5137 ‘cfv 6348 Topctop 21429 TopOnctopon 21446 Kol2ct0 21842 KQckq 22229 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-qtop 16768 df-top 21430 df-topon 21447 df-t0 21849 df-kq 22230 |
This theorem is referenced by: kqf 22283 kqhmph 22355 |
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