Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > kqt0 | Structured version Visualization version GIF version |
Description: The Kolmogorov quotient is T0 even if the original topology is not. (Contributed by Mario Carneiro, 25-Aug-2015.) |
Ref | Expression |
---|---|
kqt0 | ⊢ (𝐽 ∈ Top ↔ (KQ‘𝐽) ∈ Kol2) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2622 | . . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
2 | 1 | toptopon 20722 | . . 3 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽)) |
3 | eqid 2622 | . . . 4 ⊢ (𝑥 ∈ ∪ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}) = (𝑥 ∈ ∪ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}) | |
4 | 3 | kqt0lem 21539 | . . 3 ⊢ (𝐽 ∈ (TopOn‘∪ 𝐽) → (KQ‘𝐽) ∈ Kol2) |
5 | 2, 4 | sylbi 207 | . 2 ⊢ (𝐽 ∈ Top → (KQ‘𝐽) ∈ Kol2) |
6 | t0top 21133 | . . 3 ⊢ ((KQ‘𝐽) ∈ Kol2 → (KQ‘𝐽) ∈ Top) | |
7 | kqtop 21548 | . . 3 ⊢ (𝐽 ∈ Top ↔ (KQ‘𝐽) ∈ Top) | |
8 | 6, 7 | sylibr 224 | . 2 ⊢ ((KQ‘𝐽) ∈ Kol2 → 𝐽 ∈ Top) |
9 | 5, 8 | impbii 199 | 1 ⊢ (𝐽 ∈ Top ↔ (KQ‘𝐽) ∈ Kol2) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∈ wcel 1990 {crab 2916 ∪ cuni 4436 ↦ cmpt 4729 ‘cfv 5888 Topctop 20698 TopOnctopon 20715 Kol2ct0 21110 KQckq 21496 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-qtop 16167 df-top 20699 df-topon 20716 df-t0 21117 df-kq 21497 |
This theorem is referenced by: kqf 21550 kqhmph 21622 |
Copyright terms: Public domain | W3C validator |