![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > kqtopon | 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 |
---|---|
kqval.2 | β’ πΉ = (π₯ β π β¦ {π¦ β π½ β£ π₯ β π¦}) |
Ref | Expression |
---|---|
kqtopon | β’ (π½ β (TopOnβπ) β (KQβπ½) β (TopOnβran πΉ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | kqval.2 | . . 3 β’ πΉ = (π₯ β π β¦ {π¦ β π½ β£ π₯ β π¦}) | |
2 | 1 | kqval 23230 | . 2 β’ (π½ β (TopOnβπ) β (KQβπ½) = (π½ qTop πΉ)) |
3 | 1 | kqffn 23229 | . . . 4 β’ (π½ β (TopOnβπ) β πΉ Fn π) |
4 | dffn4 6812 | . . . 4 β’ (πΉ Fn π β πΉ:πβontoβran πΉ) | |
5 | 3, 4 | sylib 217 | . . 3 β’ (π½ β (TopOnβπ) β πΉ:πβontoβran πΉ) |
6 | qtoptopon 23208 | . . 3 β’ ((π½ β (TopOnβπ) β§ πΉ:πβontoβran πΉ) β (π½ qTop πΉ) β (TopOnβran πΉ)) | |
7 | 5, 6 | mpdan 686 | . 2 β’ (π½ β (TopOnβπ) β (π½ qTop πΉ) β (TopOnβran πΉ)) |
8 | 2, 7 | eqeltrd 2834 | 1 β’ (π½ β (TopOnβπ) β (KQβπ½) β (TopOnβran πΉ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 {crab 3433 β¦ cmpt 5232 ran crn 5678 Fn wfn 6539 βontoβwfo 6542 βcfv 6544 (class class class)co 7409 qTop cqtop 17449 TopOnctopon 22412 KQckq 23197 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7412 df-oprab 7413 df-mpo 7414 df-qtop 17453 df-top 22396 df-topon 22413 df-kq 23198 |
This theorem is referenced by: kqt0lem 23240 isr0 23241 r0cld 23242 regr1lem2 23244 kqreglem1 23245 kqreglem2 23246 kqnrmlem1 23247 kqnrmlem2 23248 kqtop 23249 |
Copyright terms: Public domain | W3C validator |