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Mirrors > Home > MPE Home > Th. List > kqtop | Structured version Visualization version GIF version |
Description: The Kolmogorov quotient is a topology on the quotient set. (Contributed by Mario Carneiro, 25-Aug-2015.) |
Ref | Expression |
---|---|
kqtop | ⊢ (𝐽 ∈ Top ↔ (KQ‘𝐽) ∈ Top) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | toptopon2 22939 | . . . 4 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽)) | |
2 | eqid 2734 | . . . . 5 ⊢ (𝑥 ∈ ∪ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}) = (𝑥 ∈ ∪ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}) | |
3 | 2 | kqtopon 23750 | . . . 4 ⊢ (𝐽 ∈ (TopOn‘∪ 𝐽) → (KQ‘𝐽) ∈ (TopOn‘ran (𝑥 ∈ ∪ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}))) |
4 | 1, 3 | sylbi 217 | . . 3 ⊢ (𝐽 ∈ Top → (KQ‘𝐽) ∈ (TopOn‘ran (𝑥 ∈ ∪ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}))) |
5 | topontop 22934 | . . 3 ⊢ ((KQ‘𝐽) ∈ (TopOn‘ran (𝑥 ∈ ∪ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})) → (KQ‘𝐽) ∈ Top) | |
6 | 4, 5 | syl 17 | . 2 ⊢ (𝐽 ∈ Top → (KQ‘𝐽) ∈ Top) |
7 | 0opn 22925 | . . . 4 ⊢ ((KQ‘𝐽) ∈ Top → ∅ ∈ (KQ‘𝐽)) | |
8 | elfvdm 6943 | . . . 4 ⊢ (∅ ∈ (KQ‘𝐽) → 𝐽 ∈ dom KQ) | |
9 | 7, 8 | syl 17 | . . 3 ⊢ ((KQ‘𝐽) ∈ Top → 𝐽 ∈ dom KQ) |
10 | ovex 7463 | . . . 4 ⊢ (𝑗 qTop (𝑥 ∈ ∪ 𝑗 ↦ {𝑦 ∈ 𝑗 ∣ 𝑥 ∈ 𝑦})) ∈ V | |
11 | df-kq 23717 | . . . 4 ⊢ KQ = (𝑗 ∈ Top ↦ (𝑗 qTop (𝑥 ∈ ∪ 𝑗 ↦ {𝑦 ∈ 𝑗 ∣ 𝑥 ∈ 𝑦}))) | |
12 | 10, 11 | dmmpti 6712 | . . 3 ⊢ dom KQ = Top |
13 | 9, 12 | eleqtrdi 2848 | . 2 ⊢ ((KQ‘𝐽) ∈ Top → 𝐽 ∈ Top) |
14 | 6, 13 | impbii 209 | 1 ⊢ (𝐽 ∈ Top ↔ (KQ‘𝐽) ∈ Top) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∈ wcel 2105 {crab 3432 ∅c0 4338 ∪ cuni 4911 ↦ cmpt 5230 dom cdm 5688 ran crn 5689 ‘cfv 6562 (class class class)co 7430 qTop cqtop 17549 Topctop 22914 TopOnctopon 22931 KQckq 23716 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-ov 7433 df-oprab 7434 df-mpo 7435 df-qtop 17553 df-top 22915 df-topon 22932 df-kq 23717 |
This theorem is referenced by: kqt0 23769 kqreg 23774 kqnrm 23775 |
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