let x, y be set ; :: thesis: for f being Function holds Im ((f +* (x .--> y)),x) = {y}
let f be Function; :: thesis: Im ((f +* (x .--> y)),x) = {y}

now :: thesis: for u being object holds
( ( u in (f +* (x .--> y)) .: {x} implies u = y ) & ( u = y implies u in (f +* (x .--> y)) .: {x} ) )

let u be object ; :: thesis: ( ( u in (f +* (x .--> y)) .: {x} implies u = y ) & ( u = y implies u in (f +* (x .--> y)) .: {x} ) )
thus ( u in (f +* (x .--> y)) .: {x} implies u = y ) :: thesis: ( u = y implies u in (f +* (x .--> y)) .: {x} )

proof

assume u in (f +* (x .--> y)) .: {x} ; :: thesis: u = y
then consider z being object such that
z in dom (f +* (x .--> y)) and
A1: z in {x} and
A2: u = (f +* (x .--> y)) . z by FUNCT_1:def 6;
z in dom (x .--> y) by A1;
then u = (x .--> y) . z by A2, FUNCT_4:13;
hence u = y by A1, FUNCOP_1:7; :: thesis: verum

end;

A3: x in {x} by TARSKI:def 1;
then A4: x in dom (x .--> y) ;
then A5: x in dom (f +* (x .--> y)) by FUNCT_4:12;
(x .--> y) . x = y by A3, FUNCOP_1:7;
then y = (f +* (x .--> y)) . x by A4, FUNCT_4:13;
hence ( u = y implies u in (f +* (x .--> y)) .: {x} ) by A3, A5, FUNCT_1:def 6; :: thesis: verum

end;

hence Im ((f +* (x .--> y)),x) = {y} by TARSKI:def 1; :: thesis: verum