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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 483 | . . . 4 β’ ((π½ β (TopOnβπ) β§ πΉ Fn π) β πΉ Fn π) | |
2 | dffn4 6820 | . . . 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 6814 | . . 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 23625 | . . . . 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 589 | . . . 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 498 | . . 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 3142 | . 2 β’ ((π½ β (TopOnβπ) β§ πΉ Fn π) β βπ₯ β (π½ qTop πΉ)(β‘πΉ β π₯) β π½) |
10 | qtoptopon 23626 | . . . 4 β’ ((π½ β (TopOnβπ) β§ πΉ:πβontoβran πΉ) β (π½ qTop πΉ) β (TopOnβran πΉ)) | |
11 | 3, 10 | syldan 589 | . . 3 β’ ((π½ β (TopOnβπ) β§ πΉ Fn π) β (π½ qTop πΉ) β (TopOnβran πΉ)) |
12 | iscn 23157 | . . 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 589 | . 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 711 | 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 394 β wcel 2098 βwral 3057 β wss 3947 β‘ccnv 5679 ran crn 5681 β cima 5683 Fn wfn 6546 βΆwf 6547 βontoβwfo 6549 βcfv 6551 (class class class)co 7424 qTop cqtop 17490 TopOnctopon 22830 Cn ccn 23146 |
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 2166 ax-ext 2698 ax-rep 5287 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-iun 5000 df-br 5151 df-opab 5213 df-mpt 5234 df-id 5578 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-ov 7427 df-oprab 7428 df-mpo 7429 df-map 8851 df-qtop 17494 df-top 22814 df-topon 22831 df-cn 23149 |
This theorem is referenced by: qtopcmplem 23629 qtopkgen 23632 qtoprest 23639 kqid 23650 qtopf1 23738 qtophmeo 23739 qustgplem 24043 circcn 33444 |
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