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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 22391 | . . . 4 β’ (π½ β Top β (intβπ½) = (π₯ β π« π β¦ βͺ (π½ β© π« π₯))) |
3 | 2 | fveq1d 6845 | . . 3 β’ (π½ β Top β ((intβπ½)βπ) = ((π₯ β π« π β¦ βͺ (π½ β© π« π₯))βπ)) |
4 | 3 | adantr 482 | . 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 2733 | . . 3 β’ (π₯ β π« π β¦ βͺ (π½ β© π« π₯)) = (π₯ β π« π β¦ βͺ (π½ β© π« π₯)) | |
6 | pweq 4575 | . . . . 5 β’ (π₯ = π β π« π₯ = π« π) | |
7 | 6 | ineq2d 4173 | . . . 4 β’ (π₯ = π β (π½ β© π« π₯) = (π½ β© π« π)) |
8 | 7 | unieqd 4880 | . . 3 β’ (π₯ = π β βͺ (π½ β© π« π₯) = βͺ (π½ β© π« π)) |
9 | 1 | topopn 22271 | . . . . 5 β’ (π½ β Top β π β π½) |
10 | elpw2g 5302 | . . . . 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 479 | . . 3 β’ ((π½ β Top β§ π β π) β π β π« π) |
13 | inex1g 5277 | . . . . 5 β’ (π½ β Top β (π½ β© π« π) β V) | |
14 | 13 | adantr 482 | . . . 4 β’ ((π½ β Top β§ π β π) β (π½ β© π« π) β V) |
15 | 14 | uniexd 7680 | . . 3 β’ ((π½ β Top β§ π β π) β βͺ (π½ β© π« π) β V) |
16 | 5, 8, 12, 15 | fvmptd3 6972 | . 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 2773 | 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 397 = wceq 1542 β wcel 2107 Vcvv 3444 β© cin 3910 β wss 3911 π« cpw 4561 βͺ cuni 4866 β¦ cmpt 5189 βcfv 6497 Topctop 22258 intcnt 22384 |
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 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-top 22259 df-ntr 22387 |
This theorem is referenced by: ntropn 22416 clsval2 22417 ntrss2 22424 ssntr 22425 isopn3 22433 ntreq0 22444 toplatglb 47112 |
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