reserve P,Q,X,Y,Z for set, p,x,x9,x1,x2,y,z for object;
reserve D for non empty set;

theorem
  for N being set, f being Function of N, bool N ex R being Relation of
  N st for i being set st i in N holds Im(R,i) = f.i
proof
  let N be set, f be Function of N, bool N;
  deffunc F(set) = f.$1;
A1: for i being Element of N st i in [#]N holds F(i) c= [#]N
  proof
    let i be Element of N;
    assume i in [#]N;
    then f.i in bool N by Th5;
    hence thesis;
  end;
  consider R being Relation of [#]N such that
A2: for i being Element of N st i in [#]N holds Im(R,i) = F(i) from
  RELSET_1:sch 3(A1);
  reconsider R as Relation of N;
  take R;
  thus thesis by A2;
end;
