![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 21523 | . . . 4 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽)) | |
2 | eqid 2798 | . . . . 5 ⊢ (𝑥 ∈ ∪ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}) = (𝑥 ∈ ∪ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}) | |
3 | 2 | kqtopon 22332 | . . . 4 ⊢ (𝐽 ∈ (TopOn‘∪ 𝐽) → (KQ‘𝐽) ∈ (TopOn‘ran (𝑥 ∈ ∪ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}))) |
4 | 1, 3 | sylbi 220 | . . 3 ⊢ (𝐽 ∈ Top → (KQ‘𝐽) ∈ (TopOn‘ran (𝑥 ∈ ∪ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}))) |
5 | topontop 21518 | . . 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 21509 | . . . 4 ⊢ ((KQ‘𝐽) ∈ Top → ∅ ∈ (KQ‘𝐽)) | |
8 | elfvdm 6677 | . . . 4 ⊢ (∅ ∈ (KQ‘𝐽) → 𝐽 ∈ dom KQ) | |
9 | 7, 8 | syl 17 | . . 3 ⊢ ((KQ‘𝐽) ∈ Top → 𝐽 ∈ dom KQ) |
10 | ovex 7168 | . . . 4 ⊢ (𝑗 qTop (𝑥 ∈ ∪ 𝑗 ↦ {𝑦 ∈ 𝑗 ∣ 𝑥 ∈ 𝑦})) ∈ V | |
11 | df-kq 22299 | . . . 4 ⊢ KQ = (𝑗 ∈ Top ↦ (𝑗 qTop (𝑥 ∈ ∪ 𝑗 ↦ {𝑦 ∈ 𝑗 ∣ 𝑥 ∈ 𝑦}))) | |
12 | 10, 11 | dmmpti 6464 | . . 3 ⊢ dom KQ = Top |
13 | 9, 12 | eleqtrdi 2900 | . 2 ⊢ ((KQ‘𝐽) ∈ Top → 𝐽 ∈ Top) |
14 | 6, 13 | impbii 212 | 1 ⊢ (𝐽 ∈ Top ↔ (KQ‘𝐽) ∈ Top) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∈ wcel 2111 {crab 3110 ∅c0 4243 ∪ cuni 4800 ↦ cmpt 5110 dom cdm 5519 ran crn 5520 ‘cfv 6324 (class class class)co 7135 qTop cqtop 16768 Topctop 21498 TopOnctopon 21515 KQckq 22298 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-qtop 16772 df-top 21499 df-topon 21516 df-kq 22299 |
This theorem is referenced by: kqt0 22351 kqreg 22356 kqnrm 22357 |
Copyright terms: Public domain | W3C validator |