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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 22278 | . . 3 β’ (π½ β (TopOnβπ) β π½ β Top) | |
2 | toponuni 22279 | . . . . . 6 β’ (π½ β (TopOnβπ) β π = βͺ π½) | |
3 | foeq2 6754 | . . . . . 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 478 | . . . 4 β’ ((π½ β (TopOnβπ) β§ πΉ:πβontoβπ) β πΉ:βͺ π½βontoβπ) |
6 | fofn 6759 | . . . 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 2733 | . . . 4 β’ βͺ π½ = βͺ π½ | |
9 | 8 | qtoptop 23067 | . . 3 β’ ((π½ β Top β§ πΉ Fn βͺ π½) β (π½ qTop πΉ) β Top) |
10 | 1, 7, 9 | syl2an2r 684 | . 2 β’ ((π½ β (TopOnβπ) β§ πΉ:πβontoβπ) β (π½ qTop πΉ) β Top) |
11 | 8 | qtopuni 23069 | . . 3 β’ ((π½ β Top β§ πΉ:βͺ π½βontoβπ) β π = βͺ (π½ qTop πΉ)) |
12 | 1, 5, 11 | syl2an2r 684 | . 2 β’ ((π½ β (TopOnβπ) β§ πΉ:πβontoβπ) β π = βͺ (π½ qTop πΉ)) |
13 | istopon 22277 | . 2 β’ ((π½ qTop πΉ) β (TopOnβπ) β ((π½ qTop πΉ) β Top β§ π = βͺ (π½ qTop πΉ))) | |
14 | 10, 12, 13 | sylanbrc 584 | 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 397 = wceq 1542 β wcel 2107 βͺ cuni 4866 Fn wfn 6492 βontoβwfo 6495 βcfv 6497 (class class class)co 7358 qTop cqtop 17390 Topctop 22258 TopOnctopon 22275 |
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 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-ov 7361 df-oprab 7362 df-mpo 7363 df-qtop 17394 df-top 22259 df-topon 22276 |
This theorem is referenced by: qtopid 23072 qtopcld 23080 qtopcn 23081 qtopeu 23083 qtoprest 23084 imastps 23088 kqtopon 23094 qtopf1 23183 qtophmeo 23184 qustgplem 23488 qtophaus 32474 |
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