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 Th51:
  x in dom(R|X) iff x in X & x in dom R
proof
  thus x in dom(R|X) implies x in X & x in dom R
  proof
    assume x in dom(R|X);
    then consider y being object such that
A1: [x,y] in R|X by XTUPLE_0:def 12;
    [x,y] in R by A1,Def9;
    hence thesis by A1,Def9,XTUPLE_0:def 12;
  end;
  assume
A2: x in X;
  assume x in dom R;
  then consider y being object such that
A3: [x,y] in R by XTUPLE_0:def 12;
  [x,y] in R|X by A2,A3,Def9;
  hence thesis by XTUPLE_0:def 12;
end;
