Power Set is Magma of Sets

Theorem
Let $X$ be a set, and let $\left\{{\phi_i: i \in I}\right\}$ be a collection of mappings.

Then $\mathcal P \left({X}\right)$, the power set of $X$, is a magma of sets for $\left\{{\phi_i: i \in I}\right\}$ on $X$.

Proof
For each $i \in I$, for each $\left({S_{j_i}}\right)_{j_i \in J_i} \in \mathcal P \left({X}\right)^{J_i} \cap \mathcal D_i$, that:


 * $\phi_i \left({\left({S_{j_i}}\right)_{j_i \in J_i}}\right) \in \mathcal P \left({X}\right)$

follows directly from the fact that $\mathcal P \left({X}\right)$ is the codomain of $\phi_i$.

Hence $\mathcal P \left({X}\right)$ is a magma of sets for $\left\{{\phi_i: i \in I}\right\}$ on $X$.