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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 21242 | . . 3 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽)) | |
2 | eqid 2772 | . . . 4 ⊢ (𝑥 ∈ ∪ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}) = (𝑥 ∈ ∪ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}) | |
3 | 2 | kqt0lem 22060 | . . 3 ⊢ (𝐽 ∈ (TopOn‘∪ 𝐽) → (KQ‘𝐽) ∈ Kol2) |
4 | 1, 3 | sylbi 209 | . 2 ⊢ (𝐽 ∈ Top → (KQ‘𝐽) ∈ Kol2) |
5 | t0top 21653 | . . 3 ⊢ ((KQ‘𝐽) ∈ Kol2 → (KQ‘𝐽) ∈ Top) | |
6 | kqtop 22069 | . . 3 ⊢ (𝐽 ∈ Top ↔ (KQ‘𝐽) ∈ Top) | |
7 | 5, 6 | sylibr 226 | . 2 ⊢ ((KQ‘𝐽) ∈ Kol2 → 𝐽 ∈ Top) |
8 | 4, 7 | impbii 201 | 1 ⊢ (𝐽 ∈ Top ↔ (KQ‘𝐽) ∈ Kol2) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∈ wcel 2050 {crab 3086 ∪ cuni 4708 ↦ cmpt 5004 ‘cfv 6185 Topctop 21217 TopOnctopon 21234 Kol2ct0 21630 KQckq 22017 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 ax-rep 5045 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-ral 3087 df-rex 3088 df-reu 3089 df-rab 3091 df-v 3411 df-sbc 3676 df-csb 3781 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4709 df-iun 4790 df-br 4926 df-opab 4988 df-mpt 5005 df-id 5308 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-ov 6977 df-oprab 6978 df-mpo 6979 df-qtop 16634 df-top 21218 df-topon 21235 df-t0 21637 df-kq 22018 |
This theorem is referenced by: kqf 22071 kqhmph 22143 |
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