Inverse Image of Set under Set-Like Relation is Set

Theorem
Let $A$ be a class.

Let $\mathcal R$ be a set-like endorelation on $A$.

Let $B \subseteq A$ be a set.

Then $\mathcal R^{-1}(B)$, the inverse image of $B$ under $\mathcal R$, is also a set.

Proof
Since $\mathcal R$ is set-like, $\mathcal R^{-1}(\{ x \})$ is a set for each $x$ in $A$.

As $B \subseteq A$, this holds also for each $x \in B$.

But then $\displaystyle \mathcal R^{-1}(B) = \bigcup_{x \in B} \mathcal R^{-1}(\{x\})$, which is a set by the Axiom of Union.