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 Th55:
  dom(R|X) = dom R /\ X
proof
  for x being object holds x in dom(R|X) iff x in dom R /\ X
  proof let x be object;
    x in dom(R|X) iff x in dom R & x in X by Th51;
    hence thesis by XBOOLE_0:def 4;
  end;
  hence thesis by TARSKI:2;
end;
