reserve Y for non empty set;
reserve B for Subset of Y;

theorem
  for a being Function of Y,BOOLEAN holds a is_dependent_of %I(Y)
proof
  let a be Function of Y,BOOLEAN;
  for F being set st F in %I(Y) holds for x1,x2 being set st x1 in F & x2
  in F holds a.x1=a.x2
  proof
    let F be set;
    assume F in %I(Y);
    then F in {B:ex z being set st B={z} & z in Y} by PARTIT1:31;
    then ex B st F=B & ex z being set st B={z} & z in Y;
    then consider z being set such that
A1: F={z} and
    z in Y;
    let x1,x2 be set;
    assume that
A2: x1 in F and
A3: x2 in F;
    x1=z by A1,A2,TARSKI:def 1;
    hence thesis by A1,A3,TARSKI:def 1;
  end;
  hence thesis;
end;
