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Mirrors > Home > MPE Home > Th. List > kqreg | Structured version Visualization version GIF version |
Description: The Kolmogorov quotient of a regular space is regular. By regr1 22355 it is also Hausdorff, so we can also say that a space is regular iff the Kolmogorov quotient is regular Hausdorff (T3). (Contributed by Mario Carneiro, 25-Aug-2015.) |
Ref | Expression |
---|---|
kqreg | ⊢ (𝐽 ∈ Reg ↔ (KQ‘𝐽) ∈ Reg) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | regtop 21938 | . . . 4 ⊢ (𝐽 ∈ Reg → 𝐽 ∈ Top) | |
2 | toptopon2 21523 | . . . 4 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽)) | |
3 | 1, 2 | sylib 221 | . . 3 ⊢ (𝐽 ∈ Reg → 𝐽 ∈ (TopOn‘∪ 𝐽)) |
4 | eqid 2798 | . . . 4 ⊢ (𝑥 ∈ ∪ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}) = (𝑥 ∈ ∪ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}) | |
5 | 4 | kqreglem1 22346 | . . 3 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝐽 ∈ Reg) → (KQ‘𝐽) ∈ Reg) |
6 | 3, 5 | mpancom 687 | . 2 ⊢ (𝐽 ∈ Reg → (KQ‘𝐽) ∈ Reg) |
7 | regtop 21938 | . . . . 5 ⊢ ((KQ‘𝐽) ∈ Reg → (KQ‘𝐽) ∈ Top) | |
8 | kqtop 22350 | . . . . 5 ⊢ (𝐽 ∈ Top ↔ (KQ‘𝐽) ∈ Top) | |
9 | 7, 8 | sylibr 237 | . . . 4 ⊢ ((KQ‘𝐽) ∈ Reg → 𝐽 ∈ Top) |
10 | 9, 2 | sylib 221 | . . 3 ⊢ ((KQ‘𝐽) ∈ Reg → 𝐽 ∈ (TopOn‘∪ 𝐽)) |
11 | 4 | kqreglem2 22347 | . . 3 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ (KQ‘𝐽) ∈ Reg) → 𝐽 ∈ Reg) |
12 | 10, 11 | mpancom 687 | . 2 ⊢ ((KQ‘𝐽) ∈ Reg → 𝐽 ∈ Reg) |
13 | 6, 12 | impbii 212 | 1 ⊢ (𝐽 ∈ Reg ↔ (KQ‘𝐽) ∈ Reg) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∈ wcel 2111 {crab 3110 ∪ cuni 4800 ↦ cmpt 5110 ‘cfv 6324 Topctop 21498 TopOnctopon 21515 Regcreg 21914 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-int 4839 df-iun 4883 df-iin 4884 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-map 8391 df-qtop 16772 df-top 21499 df-topon 21516 df-cld 21624 df-cls 21626 df-cn 21832 df-reg 21921 df-kq 22299 |
This theorem is referenced by: (None) |
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