Inverse Image of Set under Set-Like Relation is Set

Theorem

Let $A$ be a class.

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

Let $B \subseteq A$ be a set.

Then $\map {\RR^{-1} } B$, the inverse image of $B$ under $\RR$, is also a set.

Proof

Since $\RR$ is set-like, $\map {\RR^{-1} } {\set x}$ is a set for each $x$ in $A$.

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

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But then $\ds \map {\RR^{-1} } B = \bigcup_{x \mathop \in B} \map {\RR^{-1} } {\set x}$, which is a set by the Axiom of Unions.

$\blacksquare$