Preimage of Intersection under Relation/Family of Sets

Theorem

Let $S$ and $T$ be sets.

Let $\family {T_i}_{i \mathop \in I}$ be a family of subsets of $T$.

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

Then:

$\ds \RR^{-1} \sqbrk {\bigcap_{i \mathop \in I} T_i} \subseteq \bigcap_{i \mathop \in I} \RR^{-1} \sqbrk {T_i}$

where $\ds \bigcap_{i \mathop \in I} T_i$ denotes the intersection of $\family {T_i}_{i \mathop \in I}$.

Proof

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

$\blacksquare$