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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 21520 | . . . 4 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽)) | |
2 | eqid 2821 | . . . . 5 ⊢ (𝑥 ∈ ∪ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}) = (𝑥 ∈ ∪ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}) | |
3 | 2 | kqtopon 22329 | . . . 4 ⊢ (𝐽 ∈ (TopOn‘∪ 𝐽) → (KQ‘𝐽) ∈ (TopOn‘ran (𝑥 ∈ ∪ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}))) |
4 | 1, 3 | sylbi 219 | . . 3 ⊢ (𝐽 ∈ Top → (KQ‘𝐽) ∈ (TopOn‘ran (𝑥 ∈ ∪ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}))) |
5 | topontop 21515 | . . 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 21506 | . . . 4 ⊢ ((KQ‘𝐽) ∈ Top → ∅ ∈ (KQ‘𝐽)) | |
8 | elfvdm 6696 | . . . 4 ⊢ (∅ ∈ (KQ‘𝐽) → 𝐽 ∈ dom KQ) | |
9 | 7, 8 | syl 17 | . . 3 ⊢ ((KQ‘𝐽) ∈ Top → 𝐽 ∈ dom KQ) |
10 | ovex 7183 | . . . 4 ⊢ (𝑗 qTop (𝑥 ∈ ∪ 𝑗 ↦ {𝑦 ∈ 𝑗 ∣ 𝑥 ∈ 𝑦})) ∈ V | |
11 | df-kq 22296 | . . . 4 ⊢ KQ = (𝑗 ∈ Top ↦ (𝑗 qTop (𝑥 ∈ ∪ 𝑗 ↦ {𝑦 ∈ 𝑗 ∣ 𝑥 ∈ 𝑦}))) | |
12 | 10, 11 | dmmpti 6486 | . . 3 ⊢ dom KQ = Top |
13 | 9, 12 | eleqtrdi 2923 | . 2 ⊢ ((KQ‘𝐽) ∈ Top → 𝐽 ∈ Top) |
14 | 6, 13 | impbii 211 | 1 ⊢ (𝐽 ∈ Top ↔ (KQ‘𝐽) ∈ Top) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∈ wcel 2110 {crab 3142 ∅c0 4290 ∪ cuni 4831 ↦ cmpt 5138 dom cdm 5549 ran crn 5550 ‘cfv 6349 (class class class)co 7150 qTop cqtop 16770 Topctop 21495 TopOnctopon 21512 KQckq 22295 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-qtop 16774 df-top 21496 df-topon 21513 df-kq 22296 |
This theorem is referenced by: kqt0 22348 kqreg 22353 kqnrm 22354 |
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