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

now

let i be set ; :: thesis: ( i in I implies (bool (I --> {x})) . i = (I --> {{} ,{x}}) . i )
assume A1: i in I ; :: thesis: (bool (I --> {x})) . i = (I --> {{} ,{x}}) . i
hence (bool (I --> {x})) . i = bool ((I --> {x}) . i) by Def1
.= bool {x} by A1, FUNCOP_1:13
.= {{} ,{x}} by ZFMISC_1:30
.= (I --> {{} ,{x}}) . i by A1, FUNCOP_1:13 ;
:: thesis: verum

end;

hence bool (I --> {x}) = I --> {{} ,{x}} by PBOOLE:3; :: thesis: verum