Set of All Mappings is Small Class

Theorem

Let $S$ and $T$ be small classes.

It follows that the set of all mappings $S^T$ is a small class.

Proof

The set of all mappings $S^T$ is equal to the collection of all mappings $f : S \to T$.

Each of these mappings $f$ is a subset of $S \times T$.

Thus, $S^T \subseteq \powerset {S \times T}$.

Therefore, by Cartesian Product is Small and the axiom of powers, $S^T$ is a small class.

$\blacksquare$

Sources

• 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 10.48$