Operation is Isomorphic to Power Structure Operation on Set of Singleton Subsets

Theorem
Let $\struct {S, \circ}$ be a magma.

Let $\struct {\powerset S, \circ_\PP}$ be the power structure of $\struct {S, \circ}$.

Let $S'$ denote the set of singleton elements of $\powerset S$.

Then $\struct {S, \circ}$ is isomorphic to $\struct {S', \circ_\PP}$.

Proof
Let $\phi: S \to S'$ be the mapping defined as:
 * $\forall x \in S: \map \phi x = \set x$

We have that:

demonstrating that $\phi$ is an injection.

Then we have:

demonstrating that $\phi$ is a surjection.

Hence by definition $\phi$ is a bijection.

By the definition of a magma, $S$ is closed under $\circ$.

That is:
 * $\forall a, b \in S, a \circ b \in S$

Hence:
 * $a \circ b \in \Dom \phi$

Also, by Power Structure of Magma is Magma, $S'$ is closed under $\circ_\PP$.

Hence:
 * $\set a \circ_\PP \set b \in S'$

Now:

That is, $\phi$ is a homomorphism.

So $\phi$ is a bijective homomorphism.

Hence the result by definition of isomorphism.