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;

theorem Th15:
  id A c= R implies A c= dom R & A c= rng R
proof
  assume
A1: id A c= R;
  thus A c= dom R
  proof
    let x be object;
    assume x in A;
    then [x,x] in id A by RELAT_1:def 10;
    hence thesis by A1,XTUPLE_0:def 12;
  end;
  thus A c= rng R
  proof
    let x be object;
    assume x in A;
    then [x,x] in id A by RELAT_1:def 10;
    hence thesis by A1,XTUPLE_0:def 13;
  end;
end;
