Intersection of Relations is Relation

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

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

Let $\ds \RR = \bigcap \FF$, the intersection of all the elements of $\FF$.

Then $\RR$ is a relation from $S$ to $T$.

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

By Intersection of Subsets is Subset: Set of Sets:


 * $\RR \subseteq S \times T$

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