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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 23337 | . . . . . 6 ⊢ (𝐽 ∈ Kol2 → 𝐽 ∈ Top) | |
| 2 | toptopon2 22924 | . . . . . 6 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽)) | |
| 3 | 1, 2 | sylib 218 | . . . . 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 23826 | . . . . 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 267 | . . 3 ⊢ (𝐽 ∈ Kol2 → (𝑥 ∈ ∪ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}) ∈ (𝐽Homeo(KQ‘𝐽))) |
| 8 | hmphi 23785 | . . 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 23790 | . . 3 ⊢ (𝐽 ≃ (KQ‘𝐽) → (KQ‘𝐽) ≃ 𝐽) | |
| 11 | hmphtop1 23787 | . . . 4 ⊢ (𝐽 ≃ (KQ‘𝐽) → 𝐽 ∈ Top) | |
| 12 | kqt0 23754 | . . . 4 ⊢ (𝐽 ∈ Top ↔ (KQ‘𝐽) ∈ Kol2) | |
| 13 | 11, 12 | sylib 218 | . . 3 ⊢ (𝐽 ≃ (KQ‘𝐽) → (KQ‘𝐽) ∈ Kol2) |
| 14 | t0hmph 23798 | . . 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 209 | 1 ⊢ (𝐽 ∈ Kol2 ↔ 𝐽 ≃ (KQ‘𝐽)) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∈ wcel 2108 {crab 3436 ∪ cuni 4907 class class class wbr 5143 ↦ cmpt 5225 ‘cfv 6561 (class class class)co 7431 Topctop 22899 TopOnctopon 22916 Kol2ct0 23314 KQckq 23701 Homeochmeo 23761 ≃ chmph 23762 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-1o 8506 df-map 8868 df-qtop 17552 df-top 22900 df-topon 22917 df-cn 23235 df-t0 23321 df-kq 23702 df-hmeo 23763 df-hmph 23764 |
| This theorem is referenced by: ist1-5lem 23828 t1r0 23829 |
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