Minimal WRT Restriction

Theorem
Let $A$ be a set or class.

Let $\mathcal R$ be a relation on $A$.

Let $B$ be a subset or subclass of $A$.

Let $\mathcal R'$ be the restriction of $\mathcal R$ to $B$.

Let $m \in B$.

Then $m$ is $\mathcal R$-minimal in $B$ iff it is $\mathcal R'$-minimal in $B$.

Forward implication
Let $m$ be $\mathcal R$-minimal in $B$.

Let $x$ be any element of $B$.

Suppose for the sake of contradiction that $x \mathrel{\mathcal R'} m$.

Then since $\mathcal R' \subseteq \mathcal R$, $x \mathrel{\mathcal R} m$, contradicting the fact that $m$ is $\mathcal R$-minimal in $B$.

Thus $\lnot \left({x \mathrel{\mathcal R'} m}\right)$.

As this holds for all $x \in B$, $m$ is $\mathcal R'$-minimal in $B$.

Reverse implication
Let $m$ be $\mathcal R'$-minimal in $B$.

Let $x \in B$ and suppose for the sake of contradiction that $x \mathrel{\mathcal R} m$.

Then $x, m \in B$, so $(x, m) \in B \times B$.

Thus $(x,m) \in \mathcal R \cap (B \times B) = \mathcal R'$, so $x \mathrel{\mathcal R'} m$.

This contradicts the fact that $m$ is $\mathcal R'$-minimal in $B$.

Thus $\lnot \left({x \mathrel{\mathcal R} m}\right)$.

As this holds for all $x \in B$, $m$ is $\mathcal R$-minimal in $B$.