Domain of Small Relation is Small

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 Small You 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: $\S 6.8 \ (2)$