# Set of All Mappings is Small Class

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## 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, $S^T \subseteq \mathcal P \left({ S \times T }\right)$.

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*: $\S 10.48$