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Mirrors > Home > MPE Home > Th. List > restuni | Structured version Visualization version GIF version |
Description: The underlying set of a subspace topology. (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 13-Aug-2015.) |
Ref | Expression |
---|---|
restuni.1 | β’ π = βͺ π½ |
Ref | Expression |
---|---|
restuni | β’ ((π½ β Top β§ π΄ β π) β π΄ = βͺ (π½ βΎt π΄)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | restuni.1 | . . . 4 β’ π = βͺ π½ | |
2 | 1 | toptopon 22639 | . . 3 β’ (π½ β Top β π½ β (TopOnβπ)) |
3 | resttopon 22885 | . . 3 β’ ((π½ β (TopOnβπ) β§ π΄ β π) β (π½ βΎt π΄) β (TopOnβπ΄)) | |
4 | 2, 3 | sylanb 579 | . 2 β’ ((π½ β Top β§ π΄ β π) β (π½ βΎt π΄) β (TopOnβπ΄)) |
5 | toponuni 22636 | . 2 β’ ((π½ βΎt π΄) β (TopOnβπ΄) β π΄ = βͺ (π½ βΎt π΄)) | |
6 | 4, 5 | syl 17 | 1 β’ ((π½ β Top β§ π΄ β π) β π΄ = βͺ (π½ βΎt π΄)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1539 β wcel 2104 β wss 3947 βͺ cuni 4907 βcfv 6542 (class class class)co 7411 βΎt crest 17370 Topctop 22615 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-3or 1086 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-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 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-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 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-om 7858 df-1st 7977 df-2nd 7978 df-en 8942 df-fin 8945 df-fi 9408 df-rest 17372 df-topgen 17393 df-top 22616 df-topon 22633 df-bases 22669 |
This theorem is referenced by: restuni2 22891 restcld 22896 restopn2 22901 neitr 22904 restcls 22905 restntr 22906 rncmp 23120 cmpsublem 23123 cmpsub 23124 fiuncmp 23128 connsubclo 23148 connima 23149 conncn 23150 nllyrest 23210 cldllycmp 23219 lly1stc 23220 llycmpkgen2 23274 1stckgen 23278 txkgen 23376 xkopjcn 23380 xkococnlem 23383 cnextfres1 23792 cnextfres 23793 cncfcnvcn 24666 cnheibor 24701 evthicc 25208 psercn 26174 abelth 26189 zarmxt1 33158 connpconn 34524 cvmscld 34562 cvmsss2 34563 cvmliftmolem1 34570 cvmliftlem10 34583 cvmlift2lem9 34600 cvmlift2lem11 34602 cvmlift2lem12 34603 cvmlift3lem7 34614 ivthALT 35523 ptrest 36790 poimirlem29 36820 poimirlem30 36821 poimirlem31 36822 poimir 36824 cncfuni 44900 cncfiooicclem1 44907 stoweidlem28 45042 dirkercncflem4 45120 fourierdlem42 45163 restcls2lem 47632 iscnrm3rlem7 47666 |
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