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Mirrors > Home > MPE Home > Th. List > qtoptopon | Structured version Visualization version GIF version |
Description: The base set of the quotient topology. (Contributed by Mario Carneiro, 22-Aug-2015.) |
Ref | Expression |
---|---|
qtoptopon | β’ ((π½ β (TopOnβπ) β§ πΉ:πβontoβπ) β (π½ qTop πΉ) β (TopOnβπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | topontop 22833 | . . 3 β’ (π½ β (TopOnβπ) β π½ β Top) | |
2 | toponuni 22834 | . . . . . 6 β’ (π½ β (TopOnβπ) β π = βͺ π½) | |
3 | foeq2 6803 | . . . . . 6 β’ (π = βͺ π½ β (πΉ:πβontoβπ β πΉ:βͺ π½βontoβπ)) | |
4 | 2, 3 | syl 17 | . . . . 5 β’ (π½ β (TopOnβπ) β (πΉ:πβontoβπ β πΉ:βͺ π½βontoβπ)) |
5 | 4 | biimpa 475 | . . . 4 β’ ((π½ β (TopOnβπ) β§ πΉ:πβontoβπ) β πΉ:βͺ π½βontoβπ) |
6 | fofn 6808 | . . . 4 β’ (πΉ:βͺ π½βontoβπ β πΉ Fn βͺ π½) | |
7 | 5, 6 | syl 17 | . . 3 β’ ((π½ β (TopOnβπ) β§ πΉ:πβontoβπ) β πΉ Fn βͺ π½) |
8 | eqid 2725 | . . . 4 β’ βͺ π½ = βͺ π½ | |
9 | 8 | qtoptop 23622 | . . 3 β’ ((π½ β Top β§ πΉ Fn βͺ π½) β (π½ qTop πΉ) β Top) |
10 | 1, 7, 9 | syl2an2r 683 | . 2 β’ ((π½ β (TopOnβπ) β§ πΉ:πβontoβπ) β (π½ qTop πΉ) β Top) |
11 | 8 | qtopuni 23624 | . . 3 β’ ((π½ β Top β§ πΉ:βͺ π½βontoβπ) β π = βͺ (π½ qTop πΉ)) |
12 | 1, 5, 11 | syl2an2r 683 | . 2 β’ ((π½ β (TopOnβπ) β§ πΉ:πβontoβπ) β π = βͺ (π½ qTop πΉ)) |
13 | istopon 22832 | . 2 β’ ((π½ qTop πΉ) β (TopOnβπ) β ((π½ qTop πΉ) β Top β§ π = βͺ (π½ qTop πΉ))) | |
14 | 10, 12, 13 | sylanbrc 581 | 1 β’ ((π½ β (TopOnβπ) β§ πΉ:πβontoβπ) β (π½ qTop πΉ) β (TopOnβπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 394 = wceq 1533 β wcel 2098 βͺ cuni 4903 Fn wfn 6538 βontoβwfo 6541 βcfv 6543 (class class class)co 7416 qTop cqtop 17484 Topctop 22813 TopOnctopon 22830 |
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 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 |
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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7419 df-oprab 7420 df-mpo 7421 df-qtop 17488 df-top 22814 df-topon 22831 |
This theorem is referenced by: qtopid 23627 qtopcld 23635 qtopcn 23636 qtopeu 23638 qtoprest 23639 imastps 23643 kqtopon 23649 qtopf1 23738 qtophmeo 23739 qustgplem 24043 qtophaus 33494 |
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