# Domain of Small Relation is Small

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## Theorem

Let $a$ be a small class.

Let $a$ also be a relation.

Then the domain of $a$ is small.

## Proof

Let $A$ equal:

- $\left\{{\left({\left({x, y}\right), x}\right): \left({x, y}\right) \in a}\right\}$

Then, $A$ maps $a$ to its domain.

Thus, the domain of $a$ is the image of $a$ under $A$.

By Image of Small Class under Mapping is Small, the domain of $a$ is small.

$\blacksquare$

## Sources

- 1971: Gaisi Takeuti and Wilson M. Zaring:
*Introduction to Axiomatic Set Theory*: $\S 6.8 \ (2)$