A corollary to Kleene's recursion theorem states that for every Gödel numbering \phi\colon \mathbb{N} \to \mathbf{P}^{(1)} of the computable functions and every computable function Q(x,y), there is an index e such that \phi_e(y) returns Q(e,y).
Let F be a set of computable functions such that \emptyset \neq F \neq \mathbf{P}^{(1)}.