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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 | β’ (π½ β Top β (KQβπ½) β Top) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | toptopon2 22411 | . . . 4 β’ (π½ β Top β π½ β (TopOnββͺ π½)) | |
2 | eqid 2732 | . . . . 5 β’ (π₯ β βͺ π½ β¦ {π¦ β π½ β£ π₯ β π¦}) = (π₯ β βͺ π½ β¦ {π¦ β π½ β£ π₯ β π¦}) | |
3 | 2 | kqtopon 23222 | . . . 4 β’ (π½ β (TopOnββͺ π½) β (KQβπ½) β (TopOnβran (π₯ β βͺ π½ β¦ {π¦ β π½ β£ π₯ β π¦}))) |
4 | 1, 3 | sylbi 216 | . . 3 β’ (π½ β Top β (KQβπ½) β (TopOnβran (π₯ β βͺ π½ β¦ {π¦ β π½ β£ π₯ β π¦}))) |
5 | topontop 22406 | . . 3 β’ ((KQβπ½) β (TopOnβran (π₯ β βͺ π½ β¦ {π¦ β π½ β£ π₯ β π¦})) β (KQβπ½) β Top) | |
6 | 4, 5 | syl 17 | . 2 β’ (π½ β Top β (KQβπ½) β Top) |
7 | 0opn 22397 | . . . 4 β’ ((KQβπ½) β Top β β β (KQβπ½)) | |
8 | elfvdm 6925 | . . . 4 β’ (β β (KQβπ½) β π½ β dom KQ) | |
9 | 7, 8 | syl 17 | . . 3 β’ ((KQβπ½) β Top β π½ β dom KQ) |
10 | ovex 7438 | . . . 4 β’ (π qTop (π₯ β βͺ π β¦ {π¦ β π β£ π₯ β π¦})) β V | |
11 | df-kq 23189 | . . . 4 β’ KQ = (π β Top β¦ (π qTop (π₯ β βͺ π β¦ {π¦ β π β£ π₯ β π¦}))) | |
12 | 10, 11 | dmmpti 6691 | . . 3 β’ dom KQ = Top |
13 | 9, 12 | eleqtrdi 2843 | . 2 β’ ((KQβπ½) β Top β π½ β Top) |
14 | 6, 13 | impbii 208 | 1 β’ (π½ β Top β (KQβπ½) β Top) |
Colors of variables: wff setvar class |
Syntax hints: β wb 205 β wcel 2106 {crab 3432 β c0 4321 βͺ cuni 4907 β¦ cmpt 5230 dom cdm 5675 ran crn 5676 βcfv 6540 (class class class)co 7405 qTop cqtop 17445 Topctop 22386 TopOnctopon 22403 KQckq 23188 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-ov 7408 df-oprab 7409 df-mpo 7410 df-qtop 17449 df-top 22387 df-topon 22404 df-kq 23189 |
This theorem is referenced by: kqt0 23241 kqreg 23246 kqnrm 23247 |
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