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| Mirrors > Home > MPE Home > Th. List > restsn2 | Structured version Visualization version GIF version | ||
| Description: The subspace topology induced by a singleton. (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 16-Sep-2015.) |
| Ref | Expression |
|---|---|
| restsn2 | β’ ((π½ β (TopOnβπ) β§ π΄ β π) β (π½ βΎt {π΄}) = π« {π΄}) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | snssi 4811 | . . 3 β’ (π΄ β π β {π΄} β π) | |
| 2 | resttopon 22886 | . . 3 β’ ((π½ β (TopOnβπ) β§ {π΄} β π) β (π½ βΎt {π΄}) β (TopOnβ{π΄})) | |
| 3 | 1, 2 | sylan2 592 | . 2 β’ ((π½ β (TopOnβπ) β§ π΄ β π) β (π½ βΎt {π΄}) β (TopOnβ{π΄})) |
| 4 | topsn 22654 | . 2 β’ ((π½ βΎt {π΄}) β (TopOnβ{π΄}) β (π½ βΎt {π΄}) = π« {π΄}) | |
| 5 | 3, 4 | syl 17 | 1 β’ ((π½ β (TopOnβπ) β§ π΄ β π) β (π½ βΎt {π΄}) = π« {π΄}) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 395 = wceq 1540 β wcel 2105 β wss 3948 π« cpw 4602 {csn 4628 βcfv 6543 (class class class)co 7412 βΎt crest 17371 TopOnctopon 22633 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7728 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7859 df-1st 7978 df-2nd 7979 df-en 8943 df-fin 8946 df-fi 9409 df-rest 17373 df-topgen 17394 df-top 22617 df-topon 22634 df-bases 22670 |
| This theorem is referenced by: conncompid 23156 xkohaus 23378 xkoptsub 23379 cvmlift2lem9 34601 cncfdmsn 44905 |
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