# Restriction of Foundational Relation is Foundational

## Theorem

Let $S$ be a set.

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

Let $T \subseteq S$ be a subset of $S$.

Let $\mathcal R {\restriction_T} \subseteq T \times T$ be the restriction of $\mathcal R$ to $T$.

Then $\mathcal R {\restriction_T}$ is a foundational relation on $T$.

## Proof

By definition of foundational relation, every nonempty subset of $S$ has a minimal element.

By Subset Relation is Transitive, every subset of $T$ is also a subset of $S$.

Therefore every nonempty subset of $T$ has a minimal element.

Thus by definition, $\mathcal R$ is a foundational relation on $T$.

$\blacksquare$