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

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

let i be object ; :: thesis: ( i in I implies (union (I --> {x})) . i = (I --> x) . i )
assume A1: i in I ; :: thesis: (union (I --> {x})) . i = (I --> x) . i
hence (union (I --> {x})) . i = union ((I --> {x}) . i) by Def2
.= union {x} by A1, FUNCOP_1:7
.= x by ZFMISC_1:25
.= (I --> x) . i by A1, FUNCOP_1:7 ;
:: thesis: verum

end;

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