Power Set is Magma of Sets

Theorem

Let $X$ be a set.

Let $\family {\phi_i}: i \in I$ be an indexed family of mappings.

Then $\powerset X$, the power set of $X$, is a magma of sets for $\family {\phi_i}: i \in I$ on $X$.

Proof

For every $i \in I$, let $J_i$ be an index set, and let:

$\phi_i: \powerset X^{J_i} \to \powerset X$

be a partial mapping.

For each $i \in I$, for each $\family {S_{j_i} }_{j_i \mathop \in J_i} \in \powerset X^{J_i} \cap \DD_i$, that:

$\map {\phi_i} {\family {S_{j_i} }_{j_i \mathop \in J_i} } \in \powerset X$

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follows directly from the fact that $\powerset X$ is the codomain of $\phi_i$, for every indexed family $\family {S_j}_{j \mathop \in J_i} \in \SS^{J_i}$ in the domain of $\phi_i$.

Hence $\powerset X$ is a magma of sets for $\family {\phi_i}: i \in I$ on $X$.

$\blacksquare$