Preimage of Intersection under Relation/General Result

Theorem

Let $S$ and $T$ be sets.

Let $\mathcal R \subseteq S \times T$ be a relation.

Let $\mathcal P \left({T}\right)$ be the power set of $T$.

Let $\mathbb T \subseteq \mathcal P \left({T}\right)$.

Then:

$\displaystyle \mathcal R^{-1} \left[{\bigcap \mathbb T}\right] \subseteq \bigcap_{X \mathop \in \mathbb T} \mathcal R^{-1} \left[{X}\right]$

Proof

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

$\blacksquare$