Preimage of Intersection under Relation/General Result

Theorem

Let $S$ and $T$ be sets.

Let $\RR \subseteq S \times T$ be a relation.

Let $\map \PP T$ be the power set of $T$.

Let $\mathbb T \subseteq \map \PP T$.

Then:

$\ds \RR^{-1} \sqbrk{\bigcap \mathbb T} \subseteq \bigcap_{X \mathop \in \mathbb T} \RR^{-1} \sqbrk X$

Proof

This follows from Image of Intersection under Relation: General Result, and the fact that $\RR^{-1}$ is itself a relation, and therefore obeys the same rules.

$\blacksquare$