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

now

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

end;

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