Mapping whose Domain is Small Class is Small

Theorem
Let $F$ be a mapping.

Let the domain of $F$ be a small class.

Then, $F$ is a small class.

Proof
Let $A$ denote the domain of $F$.

Let $B$ denote the range of $F$.

Since $F$ is a mapping, $F$ is also a relation. Therefore:


 * $F \subseteq A \times B$ where $A \times B$ denotes the Cartesian product of $A$ and $B$.

$B$ is the image of $A$ under $F$ and is therefore a small class.

Since $A$ and $B$ are both small, their Cartesian Product is Small.

By Axiom of Subsets Equivalents, it follows that $F$ is small.