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

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

let i be object ; :: 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_1:7
.= {{},{x}} by ZFMISC_1:24
.= (I --> {{},{x}}) . i by A1, FUNCOP_1:7 ;
:: thesis: verum

end;

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