# Preimage of Intersection under Relation/Family of Sets

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## Theorem

Let $S$ and $T$ be sets.

Let $\left\langle{T_i}\right\rangle_{i \in I}$ be a family of subsets of $T$.

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

Then:

$\displaystyle \mathcal R^{-1} \left[{\bigcap_{i \mathop \in I} T_i}\right] \subseteq \bigcap_{i \mathop \in I} \mathcal R^{-1} \left[{T_i}\right]$

where $\displaystyle \bigcap_{i \mathop \in I} T_i$ denotes the intersection of $\left\langle{T_i}\right\rangle_{i \in I}$.

## Proof

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

$\blacksquare$