Restriction of Strictly Well-Founded Relation is Strictly Well-Founded

Theorem
Suppose $R$ is a foundational relation on $A$.

Let $B \subseteq A$.

It follows that $R$ is a foundational relation on $B$.

Proof
Every nonempty subset of $B$ is a nonempty subset of $A$.

Since every nonempty subset of $A$ has a minimal element, then a fortiori the nonempty subsets of $B$ must have a minimal element.

By the definition of a foundational relation, $R$ is a foundational relation on $B$.

Also see

 * Foundational Relation
 * Axiom of Foundation