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Mirrors > Home > MPE Home > Th. List > elqtop3 | Structured version Visualization version GIF version |
Description: Value of the quotient topology function. (Contributed by Mario Carneiro, 9-Apr-2015.) |
Ref | Expression |
---|---|
elqtop3 | β’ ((π½ β (TopOnβπ) β§ πΉ:πβontoβπ) β (π΄ β (π½ qTop πΉ) β (π΄ β π β§ (β‘πΉ β π΄) β π½))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | toponuni 22636 | . . . 4 β’ (π½ β (TopOnβπ) β π = βͺ π½) | |
2 | eqimss 4039 | . . . 4 β’ (π = βͺ π½ β π β βͺ π½) | |
3 | 1, 2 | syl 17 | . . 3 β’ (π½ β (TopOnβπ) β π β βͺ π½) |
4 | 3 | adantr 479 | . 2 β’ ((π½ β (TopOnβπ) β§ πΉ:πβontoβπ) β π β βͺ π½) |
5 | eqid 2730 | . . 3 β’ βͺ π½ = βͺ π½ | |
6 | 5 | elqtop 23421 | . 2 β’ ((π½ β (TopOnβπ) β§ πΉ:πβontoβπ β§ π β βͺ π½) β (π΄ β (π½ qTop πΉ) β (π΄ β π β§ (β‘πΉ β π΄) β π½))) |
7 | 4, 6 | mpd3an3 1460 | 1 β’ ((π½ β (TopOnβπ) β§ πΉ:πβontoβπ) β (π΄ β (π½ qTop πΉ) β (π΄ β π β§ (β‘πΉ β π΄) β π½))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 394 = wceq 1539 β wcel 2104 β wss 3947 βͺ cuni 4907 β‘ccnv 5674 β cima 5678 βontoβwfo 6540 βcfv 6542 (class class class)co 7411 qTop cqtop 17453 TopOnctopon 22632 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7414 df-oprab 7415 df-mpo 7416 df-qtop 17457 df-topon 22633 |
This theorem is referenced by: qtopid 23429 idqtop 23430 tgqtop 23436 qtopcld 23437 qtopcn 23438 qtopss 23439 qtoprest 23441 qtopomap 23442 kqopn 23458 qtopf1 23540 qustgpopn 23844 |
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