Inverse of Small Relation is Small

Theorem

Let $a$ be a small class.

Let $a$ also be a relation.

Then the inverse relation of $a$ is small.

Proof

Let $A$ equal:

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

Then $A$ maps $a$ to its inverse.

A specific link is needed here. In particular: $A$ in general has a name but I can't find it atm You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by searching for it, and adding it here. 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 {{LinkWanted}} from the code.

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

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

$\blacksquare$

Sources

• 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 6.8 \ (1)$