reserve p,q,x,x1,x2,y,y1,y2,z,z1,z2 for set;
reserve A,B,V,X,X1,X2,Y,Y1,Y2,Z for set;
reserve C,C1,C2,D,D1,D2 for non empty set;

theorem
  for f being Function of X,{0,1} holds f = chi(f"{1},X)
proof
  let f be Function of X,{0,1};
  now
    thus
A1: dom f = X by FUNCT_2:def 1;
    let x be object such that
A2: x in X;
    thus x in f"{1} implies f.x = 1
    proof
      assume x in f"{1};
      then f.x in {1} by FUNCT_1:def 7;
      hence thesis by TARSKI:def 1;
    end;
    assume not x in f"{1};
    then not f.x in {1} by A1,A2,FUNCT_1:def 7;
    then
A3: rng f c= {{},1} & f.x <> 1 by RELAT_1:def 19,TARSKI:def 1;
    f.x in rng f by A1,A2,FUNCT_1:def 3;
    hence f.x = {} by A3,TARSKI:def 2;
  end;
  hence thesis by Def3;
end;
