reserve A,X,X1,X2,Y,Y1,Y2 for set, a,b,c,d,x,y,z for object;
reserve P,P1,P2,Q,R,S for Relation;

theorem
  rng(R|X) = R.:X
proof
  for y being object holds y in rng(R|X) iff y in R.:X
  proof let y be object;
    thus y in rng(R|X) implies y in R.:X
    proof
      assume y in rng(R|X);
      then consider x being object such that
A1:   [x,y] in R|X by XTUPLE_0:def 13;
      x in X & [x,y] in R by A1,Def9;
      hence thesis by Def11;
    end;
    assume y in R.:X;
    then consider x such that
A2: [x,y] in R & x in X by Def11;
    [x,y] in R|X by A2,Def9;
    hence thesis by XTUPLE_0:def 13;
  end;
  hence thesis by TARSKI:2;
end;
