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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 22286 | . . . 4 β’ (π½ β (TopOnβπ) β π = βͺ π½) | |
2 | eqimss 4004 | . . . 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 482 | . 2 β’ ((π½ β (TopOnβπ) β§ πΉ:πβontoβπ) β π β βͺ π½) |
5 | eqid 2733 | . . 3 β’ βͺ π½ = βͺ π½ | |
6 | 5 | elqtop 23071 | . 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 1463 | 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 397 = wceq 1542 β wcel 2107 β wss 3914 βͺ cuni 4869 β‘ccnv 5636 β cima 5640 βontoβwfo 6498 βcfv 6500 (class class class)co 7361 qTop cqtop 17393 TopOnctopon 22282 |
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 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 |
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 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-ov 7364 df-oprab 7365 df-mpo 7366 df-qtop 17397 df-topon 22283 |
This theorem is referenced by: qtopid 23079 idqtop 23080 tgqtop 23086 qtopcld 23087 qtopcn 23088 qtopss 23089 qtoprest 23091 qtopomap 23092 kqopn 23108 qtopf1 23190 qustgpopn 23494 |
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