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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 22631 | . . . . . 6 ⊢ (𝐽 ∈ Kol2 → 𝐽 ∈ Top) | |
2 | toptopon2 22218 | . . . . . 6 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽)) | |
3 | 1, 2 | sylib 217 | . . . . 5 ⊢ (𝐽 ∈ Kol2 → 𝐽 ∈ (TopOn‘∪ 𝐽)) |
4 | eqid 2737 | . . . . . 6 ⊢ (𝑥 ∈ ∪ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}) = (𝑥 ∈ ∪ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}) | |
5 | 4 | t0kq 23120 | . . . . 5 ⊢ (𝐽 ∈ (TopOn‘∪ 𝐽) → (𝐽 ∈ Kol2 ↔ (𝑥 ∈ ∪ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}) ∈ (𝐽Homeo(KQ‘𝐽)))) |
6 | 3, 5 | syl 17 | . . . 4 ⊢ (𝐽 ∈ Kol2 → (𝐽 ∈ Kol2 ↔ (𝑥 ∈ ∪ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}) ∈ (𝐽Homeo(KQ‘𝐽)))) |
7 | 6 | ibi 266 | . . 3 ⊢ (𝐽 ∈ Kol2 → (𝑥 ∈ ∪ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}) ∈ (𝐽Homeo(KQ‘𝐽))) |
8 | hmphi 23079 | . . 3 ⊢ ((𝑥 ∈ ∪ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}) ∈ (𝐽Homeo(KQ‘𝐽)) → 𝐽 ≃ (KQ‘𝐽)) | |
9 | 7, 8 | syl 17 | . 2 ⊢ (𝐽 ∈ Kol2 → 𝐽 ≃ (KQ‘𝐽)) |
10 | hmphsym 23084 | . . 3 ⊢ (𝐽 ≃ (KQ‘𝐽) → (KQ‘𝐽) ≃ 𝐽) | |
11 | hmphtop1 23081 | . . . 4 ⊢ (𝐽 ≃ (KQ‘𝐽) → 𝐽 ∈ Top) | |
12 | kqt0 23048 | . . . 4 ⊢ (𝐽 ∈ Top ↔ (KQ‘𝐽) ∈ Kol2) | |
13 | 11, 12 | sylib 217 | . . 3 ⊢ (𝐽 ≃ (KQ‘𝐽) → (KQ‘𝐽) ∈ Kol2) |
14 | t0hmph 23092 | . . 3 ⊢ ((KQ‘𝐽) ≃ 𝐽 → ((KQ‘𝐽) ∈ Kol2 → 𝐽 ∈ Kol2)) | |
15 | 10, 13, 14 | sylc 65 | . 2 ⊢ (𝐽 ≃ (KQ‘𝐽) → 𝐽 ∈ Kol2) |
16 | 9, 15 | impbii 208 | 1 ⊢ (𝐽 ∈ Kol2 ↔ 𝐽 ≃ (KQ‘𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∈ wcel 2106 {crab 3405 ∪ cuni 4863 class class class wbr 5103 ↦ cmpt 5186 ‘cfv 6493 (class class class)co 7351 Topctop 22193 TopOnctopon 22210 Kol2ct0 22608 KQckq 22995 Homeochmeo 23055 ≃ chmph 23056 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-ov 7354 df-oprab 7355 df-mpo 7356 df-1st 7913 df-2nd 7914 df-1o 8404 df-map 8725 df-qtop 17348 df-top 22194 df-topon 22211 df-cn 22529 df-t0 22615 df-kq 22996 df-hmeo 23057 df-hmph 23058 |
This theorem is referenced by: ist1-5lem 23122 t1r0 23123 |
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