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 image 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 by Image of Small Class under Mapping is Small.

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

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

Sources

• 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 6.15$