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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 22806 | . . 3 β’ (π½ β Top β π½ β (TopOnβπ)) |
3 | resttopon 23052 | . . 3 β’ ((π½ β (TopOnβπ) β§ π΄ β π) β (π½ βΎt π΄) β (TopOnβπ΄)) | |
4 | 2, 3 | sylanb 580 | . 2 β’ ((π½ β Top β§ π΄ β π) β (π½ βΎt π΄) β (TopOnβπ΄)) |
5 | toponuni 22803 | . 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 395 = wceq 1534 β wcel 2099 β wss 3944 βͺ cuni 4903 βcfv 6542 (class class class)co 7414 βΎt crest 17393 Topctop 22782 TopOnctopon 22799 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 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 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-en 8956 df-fin 8959 df-fi 9426 df-rest 17395 df-topgen 17416 df-top 22783 df-topon 22800 df-bases 22836 |
This theorem is referenced by: restuni2 23058 restcld 23063 restopn2 23068 neitr 23071 restcls 23072 restntr 23073 rncmp 23287 cmpsublem 23290 cmpsub 23291 fiuncmp 23295 connsubclo 23315 connima 23316 conncn 23317 nllyrest 23377 cldllycmp 23386 lly1stc 23387 llycmpkgen2 23441 1stckgen 23445 txkgen 23543 xkopjcn 23547 xkococnlem 23550 cnextfres1 23959 cnextfres 23960 cncfcnvcn 24833 cnheibor 24868 evthicc 25375 psercn 26350 abelth 26365 zarmxt1 33417 connpconn 34781 cvmscld 34819 cvmsss2 34820 cvmliftmolem1 34827 cvmliftlem10 34840 cvmlift2lem9 34857 cvmlift2lem11 34859 cvmlift2lem12 34860 cvmlift3lem7 34871 ivthALT 35755 ptrest 37027 poimirlem29 37057 poimirlem30 37058 poimirlem31 37059 poimir 37061 cncfuni 45197 cncfiooicclem1 45204 stoweidlem28 45339 dirkercncflem4 45417 fourierdlem42 45460 restcls2lem 47854 iscnrm3rlem7 47888 |
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