Image of Intersection under Relation/General Result

Theorem

Let $S$ and $T$ be sets.

Let $\RR \subseteq S \times T$ be a relation.

Let $\powerset S$ be the power set of $S$.

Let $\mathbb S \subseteq \powerset S$.

Then:

$\ds \RR \sqbrk {\bigcap \mathbb S} \subseteq \bigcap_{X \mathop \in \mathbb S} \RR \sqbrk X$

$\ds \forall X \in \mathbb S: \, $

$\ds \bigcap \mathbb S$

$\subseteq$

$\ds X$

$\ds \leadsto \ \ $

$\ds \forall X \in \mathbb S: \, $

$\ds \RR \sqbrk {\bigcap \mathbb S}$

$\subseteq$

$\ds \RR \sqbrk X$

$\ds \leadsto \ \ $

$\ds \RR \sqbrk {\bigcap \mathbb S}$

$\subseteq$

$\ds \bigcap_{X \mathop \in \mathbb S} \RR \sqbrk X$

$\blacksquare$