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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 22833 | . . . 4 β’ (π½ β Top β π½ β (TopOnββͺ π½)) | |
2 | eqid 2728 | . . . . 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 23644 | . . . 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 22828 | . . 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 22819 | . . . 4 β’ ((KQβπ½) β Top β β β (KQβπ½)) | |
8 | elfvdm 6934 | . . . 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 7453 | . . . 4 β’ (π qTop (π₯ β βͺ π β¦ {π¦ β π β£ π₯ β π¦})) β V | |
11 | df-kq 23611 | . . . 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 6699 | . . 3 β’ dom KQ = Top |
13 | 9, 12 | eleqtrdi 2839 | . 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 2099 {crab 3429 β c0 4323 βͺ cuni 4908 β¦ cmpt 5231 dom cdm 5678 ran crn 5679 βcfv 6548 (class class class)co 7420 qTop cqtop 17485 Topctop 22808 TopOnctopon 22825 KQckq 23610 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-ov 7423 df-oprab 7424 df-mpo 7425 df-qtop 17489 df-top 22809 df-topon 22826 df-kq 23611 |
This theorem is referenced by: kqt0 23663 kqreg 23668 kqnrm 23669 |
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