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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 22879 | . . . 4 β’ (π½ β Top β (intβπ½) = (π₯ β π« π β¦ βͺ (π½ β© π« π₯))) |
| 3 | 2 | fveq1d 6886 | . . 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 2726 | . . 3 β’ (π₯ β π« π β¦ βͺ (π½ β© π« π₯)) = (π₯ β π« π β¦ βͺ (π½ β© π« π₯)) | |
| 6 | pweq 4611 | . . . . 5 β’ (π₯ = π β π« π₯ = π« π) | |
| 7 | 6 | ineq2d 4207 | . . . 4 β’ (π₯ = π β (π½ β© π« π₯) = (π½ β© π« π)) |
| 8 | 7 | unieqd 4915 | . . 3 β’ (π₯ = π β βͺ (π½ β© π« π₯) = βͺ (π½ β© π« π)) |
| 9 | 1 | topopn 22759 | . . . . 5 β’ (π½ β Top β π β π½) |
| 10 | elpw2g 5337 | . . . . 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 5312 | . . . . 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 7728 | . . 3 β’ ((π½ β Top β§ π β π) β βͺ (π½ β© π« π) β V) |
| 16 | 5, 8, 12, 15 | fvmptd3 7014 | . 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 2766 | 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 1533 β wcel 2098 Vcvv 3468 β© cin 3942 β wss 3943 π« cpw 4597 βͺ cuni 4902 β¦ cmpt 5224 βcfv 6536 Topctop 22746 intcnt 22872 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-top 22747 df-ntr 22875 |
| This theorem is referenced by: ntropn 22904 clsval2 22905 ntrss2 22912 ssntr 22913 isopn3 22921 ntreq0 22932 toplatglb 47881 |
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