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

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

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

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

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

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.

Now:

That is, $\phi$ is a homomorphism.

So $\phi$ is a bijective homomorphism.

Hence the result by definition of isomorphism.