Idempotent Magma Element forms Singleton Submagma

Theorem
Let $$\left({S, \circ}\right)$$ be a groupoid.

Let $$x \in S$$ be an idempotent element of $$\left({S, \circ}\right)$$.

Then $$\left({\left\{{x}\right\}, \circ}\right)$$ is a subgroupoid of $$\left({S, \circ}\right)$$.

Proof
By Singleton Subset, $$x \in S \iff \left\{{x}\right\} \subseteq S$$.

By the definiton of idempotence, $$x \circ x = x \in \left\{{x}\right\}$$.

Thus $$\left\{{x}\right\}$$ is a subset of $$S$$ which is closed under $$\circ$$.

By the definition of subgroupoid, the result follows.