let A be non empty set ; :: thesis: for y being set
for f being Function holds (f +* (A --> y)) .: A = {y}
let y be set ; :: thesis: for f being Function holds (f +* (A --> y)) .: A = {y}
let f be Function; :: thesis: (f +* (A --> y)) .: A = {y}

now

let u be set ; :: thesis: ( ( u in (f +* (A --> y)) .: A implies u = y ) & ( u = y implies u in (f +* (A --> y)) .: A ) )
thus ( u in (f +* (A --> y)) .: A implies u = y ) :: thesis: ( u = y implies u in (f +* (A --> y)) .: A )

proof

assume u in (f +* (A --> y)) .: A ; :: thesis: u = y
then consider z being set such that
A1: ( z in dom (f +* (A --> y)) & z in A & u = (f +* (A --> y)) . z ) by FUNCT_1:def 12;
z in dom (A --> y) by A1, FUNCOP_1:19;
then u = (A --> y) . z by A1, FUNCT_4:14;
hence u = y by A1, FUNCOP_1:13; :: thesis: verum

end;

consider x being set such that
A2: x in A by XBOOLE_0:def 1;
( x in dom (A --> y) & (A --> y) . x = y ) by A2, FUNCOP_1:13, FUNCOP_1:19;
then ( x in dom (f +* (A --> y)) & y = (f +* (A --> y)) . x ) by FUNCT_4:13, FUNCT_4:14;
hence ( u = y implies u in (f +* (A --> y)) .: A ) by A2, FUNCT_1:def 12; :: thesis: verum

end;

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