Metamath Proof Explorer


Theorem ntrneik4w

Description: Idempotence of the interior function is equivalent to saying a set is a neighborhood of a point if and only if the interior of the set is a neighborhood of a point. (Contributed by RP, 11-Jul-2021)

Ref Expression
Hypotheses ntrnei.o O=iV,jVk𝒫jiljmi|lkm
ntrnei.f F=𝒫BOB
ntrnei.r φIFN
Assertion ntrneik4w φs𝒫BIIs=IsxBs𝒫BsNxIsNx

Proof

Step Hyp Ref Expression
1 ntrnei.o O=iV,jVk𝒫jiljmi|lkm
2 ntrnei.f F=𝒫BOB
3 ntrnei.r φIFN
4 dfcleq Is=IIsxxIsxIIs
5 eqcom IIs=IsIs=IIs
6 ralv xVxIsxIIsxxIsxIIs
7 4 5 6 3bitr4i IIs=IsxVxIsxIIs
8 ssv BV
9 8 a1i φs𝒫BBV
10 vex xV
11 eldif xVBxV¬xB
12 10 11 mpbiran xVB¬xB
13 1 2 3 ntrneiiex φI𝒫B𝒫B
14 elmapi I𝒫B𝒫BI:𝒫B𝒫B
15 13 14 syl φI:𝒫B𝒫B
16 15 ffvelcdmda φs𝒫BIs𝒫B
17 16 elpwid φs𝒫BIsB
18 17 sseld φs𝒫BxIsxB
19 18 con3dimp φs𝒫B¬xB¬xIs
20 15 adantr φs𝒫BI:𝒫B𝒫B
21 20 16 ffvelcdmd φs𝒫BIIs𝒫B
22 21 elpwid φs𝒫BIIsB
23 22 sseld φs𝒫BxIIsxB
24 23 con3dimp φs𝒫B¬xB¬xIIs
25 19 24 2falsed φs𝒫B¬xBxIsxIIs
26 25 ex φs𝒫B¬xBxIsxIIs
27 12 26 biimtrid φs𝒫BxVBxIsxIIs
28 27 ralrimiv φs𝒫BxVBxIsxIIs
29 9 28 raldifeq φs𝒫BxBxIsxIIsxVxIsxIIs
30 3 adantr φs𝒫BIFN
31 30 adantr φs𝒫BxBIFN
32 simpr φs𝒫BxBxB
33 simplr φs𝒫BxBs𝒫B
34 1 2 31 32 33 ntrneiel φs𝒫BxBxIssNx
35 16 adantr φs𝒫BxBIs𝒫B
36 1 2 31 32 35 ntrneiel φs𝒫BxBxIIsIsNx
37 34 36 bibi12d φs𝒫BxBxIsxIIssNxIsNx
38 37 ralbidva φs𝒫BxBxIsxIIsxBsNxIsNx
39 29 38 bitr3d φs𝒫BxVxIsxIIsxBsNxIsNx
40 7 39 bitrid φs𝒫BIIs=IsxBsNxIsNx
41 40 ralbidva φs𝒫BIIs=Iss𝒫BxBsNxIsNx
42 ralcom s𝒫BxBsNxIsNxxBs𝒫BsNxIsNx
43 41 42 bitrdi φs𝒫BIIs=IsxBs𝒫BsNxIsNx