let X, Y be set ; :: thesis: for f being Function of X,Y st ( Y = {} implies X = {} ) holds
f " (f .: X) = X
let f be Function of X,Y; :: thesis: ( ( Y = {} implies X = {} ) implies f " (f .: X) = X )
assume ( Y <> {} or X = {} ) ; :: thesis: f " (f .: X) = X
then dom f = X by Def1;
then ( f " (rng f) = X & f .: X = rng f ) by RELAT_1:146, RELAT_1:169;
hence f " (f .: X) = X ; :: thesis: verum