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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 486 | . . . 4 β’ ((π½ β (TopOnβπ) β§ πΉ Fn π) β πΉ Fn π) | |
2 | dffn4 6767 | . . . 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 6761 | . . 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 23070 | . . . . 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 592 | . . . 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 501 | . . 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 3144 | . 2 β’ ((π½ β (TopOnβπ) β§ πΉ Fn π) β βπ₯ β (π½ qTop πΉ)(β‘πΉ β π₯) β π½) |
10 | qtoptopon 23071 | . . . 4 β’ ((π½ β (TopOnβπ) β§ πΉ:πβontoβran πΉ) β (π½ qTop πΉ) β (TopOnβran πΉ)) | |
11 | 3, 10 | syldan 592 | . . 3 β’ ((π½ β (TopOnβπ) β§ πΉ Fn π) β (π½ qTop πΉ) β (TopOnβran πΉ)) |
12 | iscn 22602 | . . 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 592 | . 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 712 | 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 397 β wcel 2107 βwral 3065 β wss 3915 β‘ccnv 5637 ran crn 5639 β cima 5641 Fn wfn 6496 βΆwf 6497 βontoβwfo 6499 βcfv 6501 (class class class)co 7362 qTop cqtop 17392 TopOnctopon 22275 Cn ccn 22591 |
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 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-ov 7365 df-oprab 7366 df-mpo 7367 df-map 8774 df-qtop 17396 df-top 22259 df-topon 22276 df-cn 22594 |
This theorem is referenced by: qtopcmplem 23074 qtopkgen 23077 qtoprest 23084 kqid 23095 qtopf1 23183 qtophmeo 23184 qustgplem 23488 circcn 32459 |
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