Image of Intersection under Relation/Family of Sets

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Theorem

Let $S$ and $T$ be sets.

Let $\left\langle{S_i}\right\rangle_{i \in I}$ be a family of subsets of $S$.

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


Then:

$\displaystyle \mathcal R \left[{\bigcap_{i \mathop \in I} S_i}\right] \subseteq \bigcap_{i \mathop \in I} \mathcal R \left[{S_i}\right]$

where $\displaystyle \bigcap_{i \mathop \in I} S_i$ denotes the intersection of $\left\langle{S_i}\right\rangle_{i \in I}$.


Proof

\(\displaystyle \forall i \in I: \bigcap_{j \mathop \in I} S_j\) \(\subseteq\) \(\displaystyle S_i\) $\quad$ Intersection is Subset: Family of Sets $\quad$
\(\displaystyle \implies \ \ \) \(\displaystyle \forall i \in I: \mathcal R \left[{\bigcap_{j \mathop \in I} S_j}\right]\) \(\subseteq\) \(\displaystyle \mathcal R \left[{S_i}\right]\) $\quad$ Image of Subset is Subset of Image $\quad$
\(\displaystyle \implies \ \ \) \(\displaystyle \mathcal R \left[{\bigcap_{i \mathop \in I} S_i}\right]\) \(\subseteq\) \(\displaystyle \bigcap_{i \mathop \in I} \mathcal R \left[{S_i}\right]\) $\quad$ Intersection is Largest Subset: Family of Sets $\quad$

$\blacksquare$


Sources