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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 21940 | . . . . . 6 ⊢ (𝐽 ∈ Kol2 → 𝐽 ∈ Top) | |
2 | toptopon2 21529 | . . . . . 6 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽)) | |
3 | 1, 2 | sylib 220 | . . . . 5 ⊢ (𝐽 ∈ Kol2 → 𝐽 ∈ (TopOn‘∪ 𝐽)) |
4 | eqid 2824 | . . . . . 6 ⊢ (𝑥 ∈ ∪ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}) = (𝑥 ∈ ∪ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}) | |
5 | 4 | t0kq 22429 | . . . . 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 269 | . . 3 ⊢ (𝐽 ∈ Kol2 → (𝑥 ∈ ∪ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}) ∈ (𝐽Homeo(KQ‘𝐽))) |
8 | hmphi 22388 | . . 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 22393 | . . 3 ⊢ (𝐽 ≃ (KQ‘𝐽) → (KQ‘𝐽) ≃ 𝐽) | |
11 | hmphtop1 22390 | . . . 4 ⊢ (𝐽 ≃ (KQ‘𝐽) → 𝐽 ∈ Top) | |
12 | kqt0 22357 | . . . 4 ⊢ (𝐽 ∈ Top ↔ (KQ‘𝐽) ∈ Kol2) | |
13 | 11, 12 | sylib 220 | . . 3 ⊢ (𝐽 ≃ (KQ‘𝐽) → (KQ‘𝐽) ∈ Kol2) |
14 | t0hmph 22401 | . . 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 211 | 1 ⊢ (𝐽 ∈ Kol2 ↔ 𝐽 ≃ (KQ‘𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∈ wcel 2113 {crab 3145 ∪ cuni 4841 class class class wbr 5069 ↦ cmpt 5149 ‘cfv 6358 (class class class)co 7159 Topctop 21504 TopOnctopon 21521 Kol2ct0 21917 KQckq 22304 Homeochmeo 22364 ≃ chmph 22365 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7162 df-oprab 7163 df-mpo 7164 df-1st 7692 df-2nd 7693 df-1o 8105 df-map 8411 df-qtop 16783 df-top 21505 df-topon 21522 df-cn 21838 df-t0 21924 df-kq 22305 df-hmeo 22366 df-hmph 22367 |
This theorem is referenced by: ist1-5lem 22431 t1r0 22432 |
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