On the Cardinality of the Family of Precomplete Classes in $P_E$

Let $E$ be an infinite set of cardinality $\mathbf m$, and let $P_E$ be the set of all functions defined on $E$. We prove that the cardinality of the family of all classes precomplete in $P_E$ is equal to $2^{2^{\mathbf m}}$. If $C_{\mathbb R}$ is the set of all continuous functions of real variables, then the cardinality of the family of all classes precomplete in $C_{\mathbb R}$ is equal to $2^{2^{\aleph_0}}$.