let I be set ; :: thesis: bool ([[0]] I) = I --> {{}}

now

let i be set ; :: thesis: ( i in I implies (bool ([[0]] I)) . i = (I --> {{}}) . i )
assume A1: i in I ; :: thesis: (bool ([[0]] I)) . i = (I --> {{}}) . i
then (bool ([[0]] I)) . i = bool (([[0]] I) . i) by Def1
.= {{}} by PBOOLE:5, ZFMISC_1:1 ;
hence (bool ([[0]] I)) . i = (I --> {{}}) . i by A1, FUNCOP_1:7; :: thesis: verum

end;

hence bool ([[0]] I) = I --> {{}} by PBOOLE:3; :: thesis: verum