reserve A,B,X,X1,Y,Y1,Y2,Z for set, a,x,y,z for object;
reserve P,R for Relation of X,Y;
reserve D,D1,D2,E,F for non empty set;
reserve R for Relation of D,E;
reserve x for Element of D;
reserve y for Element of E;

theorem
  for x being Element of D holds x in dom R iff ex y being Element of E
  st [x,y] in R
proof
  let x be Element of D;
  thus x in dom R implies ex y being Element of E st [x,y] in R
  proof
    assume x in dom R;
    then consider y being object such that
A1: [x,y] in R by XTUPLE_0:def 12;
    reconsider b = y as Element of E by A1,ZFMISC_1:87;
    take b;
    thus thesis by A1;
  end;
  given y being Element of E such that
A2: [x,y] in R;
  thus thesis by A2,XTUPLE_0:def 12;
end;
