Image of Intersection under Relation/General Result

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Theorem

Let $S$ and $T$ be sets.

Let $\mathcal R \subseteq S \times T$ be a relation.

Let $\mathcal P \left({S}\right)$ be the power set of $S$.

Let $\mathbb S \subseteq \mathcal P \left({S}\right)$.


Then:

$\displaystyle \mathcal R \left[{\bigcap \mathbb S}\right] \subseteq \bigcap_{X \mathop \in \mathbb S} \mathcal R \left[{X}\right]$


Proof

\(\displaystyle \forall X \in \mathbb S: \bigcap \mathbb S\) \(\subseteq\) \(\displaystyle X\) Intersection is Subset: General Result
\(\displaystyle \implies \ \ \) \(\displaystyle \forall X \in \mathbb S: \mathcal R \left[{\bigcap \mathbb S}\right]\) \(\subseteq\) \(\displaystyle \mathcal R \left[{X}\right]\) Image of Subset is Subset of Image
\(\displaystyle \implies \ \ \) \(\displaystyle \mathcal R \left[{\bigcap \mathbb S}\right]\) \(\subseteq\) \(\displaystyle \bigcap_{X \mathop \in \mathbb S} \mathcal R \left[{X}\right]\) Intersection is Largest Subset: General Result

$\blacksquare$