Power Set is Magma of Sets
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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 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:
- $\map {\phi_i} {\family {S_{j_i} }_{j_i \mathop \in J_i} } \in \powerset X$
follows directly from the fact that $\powerset X$ is the codomain of $\phi_i$.
Hence $\powerset X$ is a magma of sets for $\family {\phi_i}: i \in I$ on $X$.
$\blacksquare$