reserve x,y,z,X,Y for set;
reserve X,Y for non empty set,
  f for Function of X,Y;
reserve X, Y for non empty set,
  F for (BinOp of Y),
  B for (Element of Fin X),
  f for Function of X,Y;
reserve A for set,
  x,y,z for Element of Fin A;
reserve X,Y for non empty set,
  A for set,
  f for (Function of X, Fin A),
  i,j,k for (Element of X);

theorem Th51:
  for A being non empty set for f being Function of A, Fin A holds
  f = singleton A iff for x being Element of A holds f.x = {x}
proof
  let A be non empty set;
  let f be Function of A, Fin A;
  thus f = singleton A implies for x being Element of A holds f.x = {x} by Def6
;
  assume for x being Element of A holds f.x = {x};
  then for x be object holds x in A implies f.x = {x};
  hence thesis by Def6;
end;
