let I be set ; :: thesis: bool (EmptyMS I) = I --> {{}}

now :: thesis: for i being object st i in I holds
(bool (EmptyMS I)) . i = (I --> {{}}) . i

let i be object ; :: thesis: ( i in I implies (bool (EmptyMS I)) . i = (I --> {{}}) . i )
assume A1: i in I ; :: thesis: (bool (EmptyMS I)) . i = (I --> {{}}) . i
then (bool (EmptyMS I)) . i = bool ((EmptyMS I) . i) by Def1
.= bool {} by PBOOLE:5
.= {{}} by ZFMISC_1:1 ;
hence (bool (EmptyMS I)) . i = (I --> {{}}) . i by A1, FUNCOP_1:7; :: thesis: verum

end;

hence bool (EmptyMS I) = I --> {{}} ; :: thesis: verum