# 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

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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*: $\S 6.15$