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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 22770 | . . 3 β’ (π½ β (TopOnβπ) β π½ β Top) | |
2 | toponuni 22771 | . . . . . 6 β’ (π½ β (TopOnβπ) β π = βͺ π½) | |
3 | foeq2 6796 | . . . . . 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 476 | . . . 4 β’ ((π½ β (TopOnβπ) β§ πΉ:πβontoβπ) β πΉ:βͺ π½βontoβπ) |
6 | fofn 6801 | . . . 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 2726 | . . . 4 β’ βͺ π½ = βͺ π½ | |
9 | 8 | qtoptop 23559 | . . 3 β’ ((π½ β Top β§ πΉ Fn βͺ π½) β (π½ qTop πΉ) β Top) |
10 | 1, 7, 9 | syl2an2r 682 | . 2 β’ ((π½ β (TopOnβπ) β§ πΉ:πβontoβπ) β (π½ qTop πΉ) β Top) |
11 | 8 | qtopuni 23561 | . . 3 β’ ((π½ β Top β§ πΉ:βͺ π½βontoβπ) β π = βͺ (π½ qTop πΉ)) |
12 | 1, 5, 11 | syl2an2r 682 | . 2 β’ ((π½ β (TopOnβπ) β§ πΉ:πβontoβπ) β π = βͺ (π½ qTop πΉ)) |
13 | istopon 22769 | . 2 β’ ((π½ qTop πΉ) β (TopOnβπ) β ((π½ qTop πΉ) β Top β§ π = βͺ (π½ qTop πΉ))) | |
14 | 10, 12, 13 | sylanbrc 582 | 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 395 = wceq 1533 β wcel 2098 βͺ cuni 4902 Fn wfn 6532 βontoβwfo 6535 βcfv 6537 (class class class)co 7405 qTop cqtop 17458 Topctop 22750 TopOnctopon 22767 |
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 7722 |
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 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-ov 7408 df-oprab 7409 df-mpo 7410 df-qtop 17462 df-top 22751 df-topon 22768 |
This theorem is referenced by: qtopid 23564 qtopcld 23572 qtopcn 23573 qtopeu 23575 qtoprest 23576 imastps 23580 kqtopon 23586 qtopf1 23675 qtophmeo 23676 qustgplem 23980 qtophaus 33346 |
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