Identity Element for Power Structure

Theorem
Let $\struct {S, \circ}$ be a magma whose underlying set $S$ is non-empty.

Let $\circ_\PP$ be the operation induced on $\powerset S$, the power set of $S$.

Then:
 * a subset $J$ of $S$ is an identity element of the algebraic structure $\struct {S', \circ_\PP}$


 * there exists an identity element $e$ of $\struct {S, \circ}$, such that $J = \set e$.
 * there exists an identity element $e$ of $\struct {S, \circ}$, such that $J = \set e$.