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

now

let i be set ; :: thesis: ( i in I implies (union (I --> {{x},{y}})) . i = (I --> {x,y}) . i )
assume A1: i in I ; :: thesis: (union (I --> {{x},{y}})) . i = (I --> {x,y}) . i
hence (union (I --> {{x},{y}})) . i = union ((I --> {{x},{y}}) . i) by Def2
.= union {{x},{y}} by A1, FUNCOP_1:13
.= {x,y} by ZFMISC_1:32
.= (I --> {x,y}) . i by A1, FUNCOP_1:13 ;
:: thesis: verum

end;

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