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Mirrors > Home > MPE Home > Th. List > qtopid | Structured version Visualization version GIF version |
Description: A quotient map is a continuous function into its quotient topology. (Contributed by Mario Carneiro, 23-Mar-2015.) |
Ref | Expression |
---|---|
qtopid | β’ ((π½ β (TopOnβπ) β§ πΉ Fn π) β πΉ β (π½ Cn (π½ qTop πΉ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 484 | . . . 4 β’ ((π½ β (TopOnβπ) β§ πΉ Fn π) β πΉ Fn π) | |
2 | dffn4 6804 | . . . 4 β’ (πΉ Fn π β πΉ:πβontoβran πΉ) | |
3 | 1, 2 | sylib 217 | . . 3 β’ ((π½ β (TopOnβπ) β§ πΉ Fn π) β πΉ:πβontoβran πΉ) |
4 | fof 6798 | . . 3 β’ (πΉ:πβontoβran πΉ β πΉ:πβΆran πΉ) | |
5 | 3, 4 | syl 17 | . 2 β’ ((π½ β (TopOnβπ) β§ πΉ Fn π) β πΉ:πβΆran πΉ) |
6 | elqtop3 23558 | . . . . 5 β’ ((π½ β (TopOnβπ) β§ πΉ:πβontoβran πΉ) β (π₯ β (π½ qTop πΉ) β (π₯ β ran πΉ β§ (β‘πΉ β π₯) β π½))) | |
7 | 3, 6 | syldan 590 | . . . 4 β’ ((π½ β (TopOnβπ) β§ πΉ Fn π) β (π₯ β (π½ qTop πΉ) β (π₯ β ran πΉ β§ (β‘πΉ β π₯) β π½))) |
8 | 7 | simplbda 499 | . . 3 β’ (((π½ β (TopOnβπ) β§ πΉ Fn π) β§ π₯ β (π½ qTop πΉ)) β (β‘πΉ β π₯) β π½) |
9 | 8 | ralrimiva 3140 | . 2 β’ ((π½ β (TopOnβπ) β§ πΉ Fn π) β βπ₯ β (π½ qTop πΉ)(β‘πΉ β π₯) β π½) |
10 | qtoptopon 23559 | . . . 4 β’ ((π½ β (TopOnβπ) β§ πΉ:πβontoβran πΉ) β (π½ qTop πΉ) β (TopOnβran πΉ)) | |
11 | 3, 10 | syldan 590 | . . 3 β’ ((π½ β (TopOnβπ) β§ πΉ Fn π) β (π½ qTop πΉ) β (TopOnβran πΉ)) |
12 | iscn 23090 | . . 3 β’ ((π½ β (TopOnβπ) β§ (π½ qTop πΉ) β (TopOnβran πΉ)) β (πΉ β (π½ Cn (π½ qTop πΉ)) β (πΉ:πβΆran πΉ β§ βπ₯ β (π½ qTop πΉ)(β‘πΉ β π₯) β π½))) | |
13 | 11, 12 | syldan 590 | . 2 β’ ((π½ β (TopOnβπ) β§ πΉ Fn π) β (πΉ β (π½ Cn (π½ qTop πΉ)) β (πΉ:πβΆran πΉ β§ βπ₯ β (π½ qTop πΉ)(β‘πΉ β π₯) β π½))) |
14 | 5, 9, 13 | mpbir2and 710 | 1 β’ ((π½ β (TopOnβπ) β§ πΉ Fn π) β πΉ β (π½ Cn (π½ qTop πΉ))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 β wcel 2098 βwral 3055 β wss 3943 β‘ccnv 5668 ran crn 5670 β cima 5672 Fn wfn 6531 βΆwf 6532 βontoβwfo 6534 βcfv 6536 (class class class)co 7404 qTop cqtop 17456 TopOnctopon 22763 Cn ccn 23079 |
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 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 |
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 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7407 df-oprab 7408 df-mpo 7409 df-map 8821 df-qtop 17460 df-top 22747 df-topon 22764 df-cn 23082 |
This theorem is referenced by: qtopcmplem 23562 qtopkgen 23565 qtoprest 23572 kqid 23583 qtopf1 23671 qtophmeo 23672 qustgplem 23976 circcn 33348 |
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