Image of Intersection under One-to-Many Relation/General Result

Theorem
Let $S$ and $T$ be sets.

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

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

Then:
 * $\displaystyle \forall \mathbb S \subseteq \powerset S: \mathcal R \sqbrk {\bigcap \mathbb S} = \bigcap_{X \mathop \in \mathbb S} \mathcal R \sqbrk X$

$\mathcal R$ is one-to-many.

Sufficient Condition
Suppose:
 * $\displaystyle \mathcal R \sqbrk {\bigcap \mathbb S} = \bigcap_{X \mathop \in \mathbb S} \mathcal R \sqbrk X$

where $\mathbb S$ is any subset of $\powerset S$.

Then by definition of $\mathbb S$:
 * $\forall S_1, S_2 \in \mathbb S: \mathcal R \sqbrk {S_1 \cap S_2} = \mathcal R \sqbrk {S_1} \cap \mathcal R \sqbrk {S_2}$

and the sufficient condition applies for Image of Intersection under One-to-Many Relation.

So $\mathcal R$ is one-to-many.

Necessary Condition
Suppose $\mathcal R$ is one-to-many.

From Image of Intersection under Relation: General Result, we already have:
 * $\displaystyle \mathcal R \sqbrk {\bigcap \mathbb S} \subseteq \bigcap_{X \mathop \in \mathbb S} \mathcal R \sqbrk X$

so we just need to show:
 * $\displaystyle \forall \mathbb S \subseteq \powerset S: \bigcap_{X \mathop \in \mathbb S} \mathcal R \sqbrk X \subseteq \mathcal R \sqbrk {\bigcap \mathbb S}$

Let:
 * $\displaystyle t \in \bigcap_{X \mathop \in \mathbb S} \mathcal R \sqbrk X$

Then:

So if $\mathcal R$ is one-to-many, it follows that:
 * $\displaystyle \forall \mathbb S \subseteq \powerset S: \mathcal R \sqbrk {\bigcap \mathbb S} = \bigcap_{X \mathop \in \mathbb S} \mathcal R \sqbrk X$

Putting the results together:

$\mathcal R$ is one-to-many :
 * $\displaystyle \mathcal R \sqbrk {\bigcap \mathbb S} = \bigcap_{X \mathop \in \mathbb S} \mathcal R \sqbrk X$

where $\mathbb S$ is any subset of $\powerset S$.