Intersection of Magmas of Sets is Magma of Sets

Theorem

Let $X$ be a set.

Let $\Phi := \set {\phi_i: i \in I}$ be a collection of partial mappings with codomain $\powerset X$, the power set of $X$.

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

Then:

$\SS := \ds \bigcap_{j \mathop \in J} \SS_j$

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

Proof

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

$\map {\phi_i} {\sequence {S_{j, j_i} }_{j_i \mathop \in J_i} } \in \SS_j$

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

$(1): \quad \sequence {S_{j_i} }_{j_i \mathop \in J_i} \in \SS_j^{J_i}$

it follows from definition of set intersection that:

$\map {\phi_i} {\sequence {S_{j_i} }_{j_i \mathop \in J_i} } \in \SS$

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

$\sequence {S_{j_i} }_{j_i \mathop \in J_i} \in \SS^{J_i}$

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

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

$\blacksquare$