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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 21931 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 21515 | . . . 4 ⊢ (𝐽 ∈ Reg → 𝐽 ∈ Top) | |
2 | eqid 2825 | . . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
3 | 2 | toptopon 21099 | . . . 4 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽)) |
4 | 1, 3 | sylib 210 | . . 3 ⊢ (𝐽 ∈ Reg → 𝐽 ∈ (TopOn‘∪ 𝐽)) |
5 | eqid 2825 | . . . 4 ⊢ (𝑥 ∈ ∪ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}) = (𝑥 ∈ ∪ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}) | |
6 | 5 | kqreglem1 21922 | . . 3 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝐽 ∈ Reg) → (KQ‘𝐽) ∈ Reg) |
7 | 4, 6 | mpancom 679 | . 2 ⊢ (𝐽 ∈ Reg → (KQ‘𝐽) ∈ Reg) |
8 | regtop 21515 | . . . . 5 ⊢ ((KQ‘𝐽) ∈ Reg → (KQ‘𝐽) ∈ Top) | |
9 | kqtop 21926 | . . . . 5 ⊢ (𝐽 ∈ Top ↔ (KQ‘𝐽) ∈ Top) | |
10 | 8, 9 | sylibr 226 | . . . 4 ⊢ ((KQ‘𝐽) ∈ Reg → 𝐽 ∈ Top) |
11 | 10, 3 | sylib 210 | . . 3 ⊢ ((KQ‘𝐽) ∈ Reg → 𝐽 ∈ (TopOn‘∪ 𝐽)) |
12 | 5 | kqreglem2 21923 | . . 3 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ (KQ‘𝐽) ∈ Reg) → 𝐽 ∈ Reg) |
13 | 11, 12 | mpancom 679 | . 2 ⊢ ((KQ‘𝐽) ∈ Reg → 𝐽 ∈ Reg) |
14 | 7, 13 | impbii 201 | 1 ⊢ (𝐽 ∈ Reg ↔ (KQ‘𝐽) ∈ Reg) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∈ wcel 2164 {crab 3121 ∪ cuni 4660 ↦ cmpt 4954 ‘cfv 6127 Topctop 21075 TopOnctopon 21092 Regcreg 21491 KQckq 21874 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-int 4700 df-iun 4744 df-iin 4745 df-br 4876 df-opab 4938 df-mpt 4955 df-id 5252 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-map 8129 df-qtop 16527 df-top 21076 df-topon 21093 df-cld 21201 df-cls 21203 df-cn 21409 df-reg 21498 df-kq 21875 |
This theorem is referenced by: (None) |
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