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| Mirrors > Home > MPE Home > Th. List > ntrval | Structured version Visualization version GIF version | ||
| Description: The interior of a subset of a topology's base set is the union of all the open sets it includes. Definition of interior of [Munkres] p. 94. (Contributed by NM, 10-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.) |
| Ref | Expression |
|---|---|
| iscld.1 | β’ π = βͺ π½ |
| Ref | Expression |
|---|---|
| ntrval | β’ ((π½ β Top β§ π β π) β ((intβπ½)βπ) = βͺ (π½ β© π« π)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | iscld.1 | . . . . 5 β’ π = βͺ π½ | |
| 2 | 1 | ntrfval 22941 | . . . 4 β’ (π½ β Top β (intβπ½) = (π₯ β π« π β¦ βͺ (π½ β© π« π₯))) |
| 3 | 2 | fveq1d 6899 | . . 3 β’ (π½ β Top β ((intβπ½)βπ) = ((π₯ β π« π β¦ βͺ (π½ β© π« π₯))βπ)) |
| 4 | 3 | adantr 480 | . 2 β’ ((π½ β Top β§ π β π) β ((intβπ½)βπ) = ((π₯ β π« π β¦ βͺ (π½ β© π« π₯))βπ)) |
| 5 | eqid 2728 | . . 3 β’ (π₯ β π« π β¦ βͺ (π½ β© π« π₯)) = (π₯ β π« π β¦ βͺ (π½ β© π« π₯)) | |
| 6 | pweq 4617 | . . . . 5 β’ (π₯ = π β π« π₯ = π« π) | |
| 7 | 6 | ineq2d 4212 | . . . 4 β’ (π₯ = π β (π½ β© π« π₯) = (π½ β© π« π)) |
| 8 | 7 | unieqd 4921 | . . 3 β’ (π₯ = π β βͺ (π½ β© π« π₯) = βͺ (π½ β© π« π)) |
| 9 | 1 | topopn 22821 | . . . . 5 β’ (π½ β Top β π β π½) |
| 10 | elpw2g 5346 | . . . . 5 β’ (π β π½ β (π β π« π β π β π)) | |
| 11 | 9, 10 | syl 17 | . . . 4 β’ (π½ β Top β (π β π« π β π β π)) |
| 12 | 11 | biimpar 477 | . . 3 β’ ((π½ β Top β§ π β π) β π β π« π) |
| 13 | inex1g 5319 | . . . . 5 β’ (π½ β Top β (π½ β© π« π) β V) | |
| 14 | 13 | adantr 480 | . . . 4 β’ ((π½ β Top β§ π β π) β (π½ β© π« π) β V) |
| 15 | 14 | uniexd 7747 | . . 3 β’ ((π½ β Top β§ π β π) β βͺ (π½ β© π« π) β V) |
| 16 | 5, 8, 12, 15 | fvmptd3 7028 | . 2 β’ ((π½ β Top β§ π β π) β ((π₯ β π« π β¦ βͺ (π½ β© π« π₯))βπ) = βͺ (π½ β© π« π)) |
| 17 | 4, 16 | eqtrd 2768 | 1 β’ ((π½ β Top β§ π β π) β ((intβπ½)βπ) = βͺ (π½ β© π« π)) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β wb 205 β§ wa 395 = wceq 1534 β wcel 2099 Vcvv 3471 β© cin 3946 β wss 3947 π« cpw 4603 βͺ cuni 4908 β¦ cmpt 5231 βcfv 6548 Topctop 22808 intcnt 22934 |
| 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 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-top 22809 df-ntr 22937 |
| This theorem is referenced by: ntropn 22966 clsval2 22967 ntrss2 22974 ssntr 22975 isopn3 22983 ntreq0 22994 toplatglb 48012 |
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