# 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:

- $\set {\tuple {\tuple {x, y}, x}: \tuple {x, y} \in a}$

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.

This article, or a section of it, needs explaining.In particular: And exactly why is $A$ small, so that this result applies? Same qn for Range of Small Relation is SmallYou can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Explain}}` from the code. |

$\blacksquare$

## Sources

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