Set of All Mappings is Small Class

From ProofWiki
Jump to navigation Jump to search

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