
theorem
  for X being set, R being Relation of X
  holds R is total reflexive iff id X c= R
proof
  let X be set, R be Relation of X;
  hereby
    assume R is total reflexive;
    then A1: dom R = X & id field R c= R by PARTFUN1:def 2, RELAT_2:1;
    then field R = X \/ rng R by RELAT_1:def 6;
    hence id X c= R by A1, XBOOLE_1:12;
  end;
  assume A2: id X c= R;
  field R c= X \/ X by RELSET_1:8;
  then id field R c= id X by FUNCT_4:3;
  then A3: id field R c= R by A2;
  dom id X c= dom R by A2, RELAT_1:11;
  hence thesis by A3, XBOOLE_0:def 10, RELAT_2:1, PARTFUN1:def 2;
end;
