Intersection of Magmas of Sets is Magma of Sets

Theorem
Let $X$ be a set, and let $\Phi := \left\{{\phi_i: i \in I}\right\}$ be a collection of partial mappings with codomain $\mathcal P \left({X}\right)$, the power set of $X$.

Let $\mathcal S_j$ be a magma of sets for $\Phi$, for each $j \in J$, for some index set $J$.

Then:


 * $\mathcal S := \displaystyle \bigcap_{j \mathop \in J} \mathcal S_j$

is also a magma of sets for $\Phi$.

Proof
For each $i \in I$ and $j \in J$, we have:


 * $\phi_i \left({\left({S_{j, j_i}\right)_{j_i \in J_i}}\right) \in \mathcal S_j$

Thus, if for each $j \in J$, one has:


 * $(1): \quad \left({S_{j_i}\right)_{j_i \in J_i} \in \mathcal S_j^{J_i}$

it follows from definition of set intersection that:


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

The condition $(1)$, for each $j \in J$, comes down to:


 * $\left({S_{j_i}\right)_{j_i \in J_i} \in \mathcal S^{J_i}$

It follows that $\mathcal S$ satisfies the requirement on $\phi_i$ to be a magma of sets, for each $i \in I$.

That is to say, $\mathcal S$ is a magma of sets for $\Phi$.