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Mirrors > Home > MPE Home > Th. List > kqhmph | Structured version Visualization version GIF version |
Description: A topological space is T0 iff it is homeomorphic to its Kolmogorov quotient. (Contributed by Mario Carneiro, 25-Aug-2015.) |
Ref | Expression |
---|---|
kqhmph | ⊢ (𝐽 ∈ Kol2 ↔ 𝐽 ≃ (KQ‘𝐽)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | t0top 20943 | . . . . . 6 ⊢ (𝐽 ∈ Kol2 → 𝐽 ∈ Top) | |
2 | eqid 2610 | . . . . . . 7 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
3 | 2 | toptopon 20548 | . . . . . 6 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽)) |
4 | 1, 3 | sylib 207 | . . . . 5 ⊢ (𝐽 ∈ Kol2 → 𝐽 ∈ (TopOn‘∪ 𝐽)) |
5 | eqid 2610 | . . . . . 6 ⊢ (𝑥 ∈ ∪ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}) = (𝑥 ∈ ∪ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}) | |
6 | 5 | t0kq 21431 | . . . . 5 ⊢ (𝐽 ∈ (TopOn‘∪ 𝐽) → (𝐽 ∈ Kol2 ↔ (𝑥 ∈ ∪ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}) ∈ (𝐽Homeo(KQ‘𝐽)))) |
7 | 4, 6 | syl 17 | . . . 4 ⊢ (𝐽 ∈ Kol2 → (𝐽 ∈ Kol2 ↔ (𝑥 ∈ ∪ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}) ∈ (𝐽Homeo(KQ‘𝐽)))) |
8 | 7 | ibi 255 | . . 3 ⊢ (𝐽 ∈ Kol2 → (𝑥 ∈ ∪ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}) ∈ (𝐽Homeo(KQ‘𝐽))) |
9 | hmphi 21390 | . . 3 ⊢ ((𝑥 ∈ ∪ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}) ∈ (𝐽Homeo(KQ‘𝐽)) → 𝐽 ≃ (KQ‘𝐽)) | |
10 | 8, 9 | syl 17 | . 2 ⊢ (𝐽 ∈ Kol2 → 𝐽 ≃ (KQ‘𝐽)) |
11 | hmphsym 21395 | . . 3 ⊢ (𝐽 ≃ (KQ‘𝐽) → (KQ‘𝐽) ≃ 𝐽) | |
12 | hmphtop1 21392 | . . . 4 ⊢ (𝐽 ≃ (KQ‘𝐽) → 𝐽 ∈ Top) | |
13 | kqt0 21359 | . . . 4 ⊢ (𝐽 ∈ Top ↔ (KQ‘𝐽) ∈ Kol2) | |
14 | 12, 13 | sylib 207 | . . 3 ⊢ (𝐽 ≃ (KQ‘𝐽) → (KQ‘𝐽) ∈ Kol2) |
15 | t0hmph 21403 | . . 3 ⊢ ((KQ‘𝐽) ≃ 𝐽 → ((KQ‘𝐽) ∈ Kol2 → 𝐽 ∈ Kol2)) | |
16 | 11, 14, 15 | sylc 63 | . 2 ⊢ (𝐽 ≃ (KQ‘𝐽) → 𝐽 ∈ Kol2) |
17 | 10, 16 | impbii 198 | 1 ⊢ (𝐽 ∈ Kol2 ↔ 𝐽 ≃ (KQ‘𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∈ wcel 1977 {crab 2900 ∪ cuni 4372 class class class wbr 4583 ↦ cmpt 4643 ‘cfv 5804 (class class class)co 6549 Topctop 20517 TopOnctopon 20518 Kol2ct0 20920 KQckq 21306 Homeochmeo 21366 ≃ chmph 21367 |
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-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-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-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 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-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-1st 7059 df-2nd 7060 df-1o 7447 df-map 7746 df-qtop 15990 df-top 20521 df-topon 20523 df-cn 20841 df-t0 20927 df-kq 21307 df-hmeo 21368 df-hmph 21369 |
This theorem is referenced by: ist1-5lem 21433 t1r0 21434 |
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