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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 22764 | . . . 4 β’ (π½ β Top β π½ β (TopOnββͺ π½)) | |
2 | eqid 2724 | . . . . 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 23575 | . . . 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 22759 | . . 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 22750 | . . . 4 β’ ((KQβπ½) β Top β β β (KQβπ½)) | |
8 | elfvdm 6919 | . . . 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 7435 | . . . 4 β’ (π qTop (π₯ β βͺ π β¦ {π¦ β π β£ π₯ β π¦})) β V | |
11 | df-kq 23542 | . . . 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 6685 | . . 3 β’ dom KQ = Top |
13 | 9, 12 | eleqtrdi 2835 | . 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 2098 {crab 3424 β c0 4315 βͺ cuni 4900 β¦ cmpt 5222 dom cdm 5667 ran crn 5668 βcfv 6534 (class class class)co 7402 qTop cqtop 17454 Topctop 22739 TopOnctopon 22756 KQckq 23541 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-ov 7405 df-oprab 7406 df-mpo 7407 df-qtop 17458 df-top 22740 df-topon 22757 df-kq 23542 |
This theorem is referenced by: kqt0 23594 kqreg 23599 kqnrm 23600 |
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