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
  x in dom R iff Im(R,x) <> {}
proof
  thus x in dom R implies Im(R,x) <> {}
  proof
    assume x in dom R;
    then ex y being object st [x,y] in R by XTUPLE_0:def 12;
    hence Im(R,x) <> {} by Th159;
  end;
  assume Im(R,x) <> {};
  then consider y being object such that
A1: y in Im(R,x) by XBOOLE_0:def 1;
  [x,y] in R by A1,Th159;
  hence x in dom R by XTUPLE_0:def 12;
end;
