Restriction of Foundational Relation is Foundational

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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 $T$.

$\blacksquare$


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