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

let u be set ; :: 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 set such that
z in dom (f +* (x .--> y)) and
A1: z in {x} and
A2: u = (f +* (x .--> y)) . z by FUNCT_1:def 12;
z in dom (x .--> y) by A1, FUNCOP_1:19;
then u = (x .--> y) . z by A2, FUNCT_4:14;
hence u = y by A1, FUNCOP_1:13; :: thesis: verum

end;

A3: x in {x} by TARSKI:def 1;
then A4: x in dom (x .--> y) by FUNCOP_1:19;
then A5: x in dom (f +* (x .--> y)) by FUNCT_4:13;
(x .--> y) . x = y by A3, FUNCOP_1:13;
then y = (f +* (x .--> y)) . x by A4, FUNCT_4:14;
hence ( u = y implies u in (f +* (x .--> y)) .: {x} ) by A3, A5, FUNCT_1:def 12; :: thesis: verum

end;

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