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_1:13; :: thesis: verum

end;

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