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Mirrors > Home > ILE Home > Th. List > ntrval | Unicode 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 |
Ref | Expression |
---|---|
ntrval |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iscld.1 | . . . . 5 | |
2 | 1 | ntrfval 12269 | . . . 4 |
3 | 2 | fveq1d 5423 | . . 3 |
4 | 3 | adantr 274 | . 2 |
5 | eqid 2139 | . . 3 | |
6 | pweq 3513 | . . . . 5 | |
7 | 6 | ineq2d 3277 | . . . 4 |
8 | 7 | unieqd 3747 | . . 3 |
9 | 1 | topopn 12175 | . . . . 5 |
10 | elpw2g 4081 | . . . . 5 | |
11 | 9, 10 | syl 14 | . . . 4 |
12 | 11 | biimpar 295 | . . 3 |
13 | inex1g 4064 | . . . . 5 | |
14 | 13 | adantr 274 | . . . 4 |
15 | uniexg 4361 | . . . 4 | |
16 | 14, 15 | syl 14 | . . 3 |
17 | 5, 8, 12, 16 | fvmptd3 5514 | . 2 |
18 | 4, 17 | eqtrd 2172 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wcel 1480 cvv 2686 cin 3070 wss 3071 cpw 3510 cuni 3736 cmpt 3989 cfv 5123 ctop 12164 cnt 12262 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-top 12165 df-ntr 12265 |
This theorem is referenced by: ntropn 12286 ntrss 12288 ntrss2 12290 ssntr 12291 isopn3 12294 ntreq0 12301 |
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