reserve P,Q,X,Y,Z for set, p,x,x9,x1,x2,y,z for object;
reserve D for non empty set;
reserve A,B for non empty set;

theorem
  for X,Y,Z being set, D being non empty set, f being Function of X,D st
  Y c= X & f.:Y c= Z holds f|Y is Function of Y,Z
proof
  let X,Y,Z be set, D be non empty set, f be Function of X,D;
  assume that
A1: Y c= X and
A2: f.:Y c= Z;
  dom f = X by Def1;
  then
A3: dom(f|Y) = Y by A1,RELAT_1:62;
A4: now
    assume Z = {};
    then rng(f|Y) = {} by A2,RELAT_1:115;
    hence Y = {} by A3,RELAT_1:42;
  end;
  rng(f|Y) c= Z by A2,RELAT_1:115;
  hence thesis by A3,A4,Def1,RELSET_1:4;
end;
