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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 22360 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 21943 | . . . 4 ⊢ (𝐽 ∈ Reg → 𝐽 ∈ Top) | |
2 | toptopon2 21528 | . . . 4 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽)) | |
3 | 1, 2 | sylib 220 | . . 3 ⊢ (𝐽 ∈ Reg → 𝐽 ∈ (TopOn‘∪ 𝐽)) |
4 | eqid 2823 | . . . 4 ⊢ (𝑥 ∈ ∪ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}) = (𝑥 ∈ ∪ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}) | |
5 | 4 | kqreglem1 22351 | . . 3 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝐽 ∈ Reg) → (KQ‘𝐽) ∈ Reg) |
6 | 3, 5 | mpancom 686 | . 2 ⊢ (𝐽 ∈ Reg → (KQ‘𝐽) ∈ Reg) |
7 | regtop 21943 | . . . . 5 ⊢ ((KQ‘𝐽) ∈ Reg → (KQ‘𝐽) ∈ Top) | |
8 | kqtop 22355 | . . . . 5 ⊢ (𝐽 ∈ Top ↔ (KQ‘𝐽) ∈ Top) | |
9 | 7, 8 | sylibr 236 | . . . 4 ⊢ ((KQ‘𝐽) ∈ Reg → 𝐽 ∈ Top) |
10 | 9, 2 | sylib 220 | . . 3 ⊢ ((KQ‘𝐽) ∈ Reg → 𝐽 ∈ (TopOn‘∪ 𝐽)) |
11 | 4 | kqreglem2 22352 | . . 3 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ (KQ‘𝐽) ∈ Reg) → 𝐽 ∈ Reg) |
12 | 10, 11 | mpancom 686 | . 2 ⊢ ((KQ‘𝐽) ∈ Reg → 𝐽 ∈ Reg) |
13 | 6, 12 | impbii 211 | 1 ⊢ (𝐽 ∈ Reg ↔ (KQ‘𝐽) ∈ Reg) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∈ wcel 2114 {crab 3144 ∪ cuni 4840 ↦ cmpt 5148 ‘cfv 6357 Topctop 21503 TopOnctopon 21520 Regcreg 21919 KQckq 22303 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-map 8410 df-qtop 16782 df-top 21504 df-topon 21521 df-cld 21629 df-cls 21631 df-cn 21837 df-reg 21926 df-kq 22304 |
This theorem is referenced by: (None) |
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