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
  R.:X = R.:(dom R /\ X)
proof
  for y being object holds y in R.:X iff y in R.:(dom R /\ X)
  proof let y be object;
    thus y in R.:(X) implies y in R.:(dom R /\ X)
    proof
      assume y in R.:(X);
      then consider x such that
A1:   x in dom R and
A2:   [x,y] in R and
A3:   x in X by Th104;
      x in dom R /\ X by A1,A3,XBOOLE_0:def 4;
      hence thesis by A2,Def11;
    end;
    assume y in R.:(dom R /\ X);
    then consider x such that
    x in dom R and
A4: [x,y] in R and
A5: x in dom R /\ X by Th104;
    x in X by A5,XBOOLE_0:def 4;
    hence thesis by A4,Def11;
  end;
  hence thesis by TARSKI:2;
end;
