Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 22194 | . . 3 β’ (π½ β Top β π½ β (TopOnβπ)) |
3 | resttopon 22440 | . . 3 β’ ((π½ β (TopOnβπ) β§ π΄ β π) β (π½ βΎt π΄) β (TopOnβπ΄)) | |
4 | 2, 3 | sylanb 582 | . 2 β’ ((π½ β Top β§ π΄ β π) β (π½ βΎt π΄) β (TopOnβπ΄)) |
5 | toponuni 22191 | . 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 397 = wceq 1542 β wcel 2107 β wss 3909 βͺ cuni 4864 βcfv 6492 (class class class)co 7350 βΎt crest 17238 Topctop 22170 TopOnctopon 22187 |
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 2709 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7663 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-ral 3064 df-rex 3073 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-int 4907 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6444 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-ov 7353 df-oprab 7354 df-mpo 7355 df-om 7794 df-1st 7912 df-2nd 7913 df-en 8818 df-fin 8821 df-fi 9281 df-rest 17240 df-topgen 17261 df-top 22171 df-topon 22188 df-bases 22224 |
This theorem is referenced by: restuni2 22446 restcld 22451 restopn2 22456 neitr 22459 restcls 22460 restntr 22461 rncmp 22675 cmpsublem 22678 cmpsub 22679 fiuncmp 22683 connsubclo 22703 connima 22704 conncn 22705 nllyrest 22765 cldllycmp 22774 lly1stc 22775 llycmpkgen2 22829 1stckgen 22833 txkgen 22931 xkopjcn 22935 xkococnlem 22938 cnextfres1 23347 cnextfres 23348 cncfcnvcn 24216 cnheibor 24246 evthicc 24751 psercn 25713 abelth 25728 zarmxt1 32241 connpconn 33609 cvmscld 33647 cvmsss2 33648 cvmliftmolem1 33655 cvmliftlem10 33668 cvmlift2lem9 33685 cvmlift2lem11 33687 cvmlift2lem12 33688 cvmlift3lem7 33699 ivthALT 34738 ptrest 36008 poimirlem29 36038 poimirlem30 36039 poimirlem31 36040 poimir 36042 cncfuni 43918 cncfiooicclem1 43925 stoweidlem28 44060 dirkercncflem4 44138 fourierdlem42 44181 restcls2lem 46736 iscnrm3rlem7 46770 |
Copyright terms: Public domain | W3C validator |