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

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

let i be object ; :: 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_1:7
.= {x,y} by ZFMISC_1:26
.= (I --> {x,y}) . i by A1, FUNCOP_1:7 ;
:: thesis: verum

end;

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