Intersection of Relations is Relation

Theorem
Let $S$ and $T$ be sets.

Let $\mathcal F$ be a family of relations from $S$ to $T$.

Let $\displaystyle \mathcal R = \bigcap \mathcal F$, the intersection of all the elements of $\mathcal F$.

Then $\mathcal R$ is a relation from $S$ to $T$.

Proof
By the definition of a relation from $S$ to $T$, each element of $\mathcal F$ is a subset of $S \times T$.

By Intersection of Subsets is Subset: Set of Sets:


 * $\mathcal R \subseteq S \times T$

Therefore, by the definition of a relation from $S$ to $T$, $\mathcal R$ is a relation from $S$ to $T$.