Preimage of Intersection under Mapping

Theorem
Let $f: S \to T$ be a mapping.

Let $T_1$ and $T_2$ be subsets of $T$.

Then:
 * $f^{-1} \left({T_1 \cap T_2}\right) = f^{-1} \left({T_1}\right) \cap f^{-1} \left({T_2}\right)$.

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

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

Then:
 * $\displaystyle f \left({\bigcap \mathbb T}\right) = \bigcap_{X \in \mathbb T} f \left({X}\right)$

Proof
As $f$, being a mapping, is also a many-to-one relation, it follows from Inverse of Many-to-One Relation is One-to-Many that its inverse $f^{-1}$ is a one-to-many relation.

Thus we can apply One-to-Many Image of Intersections:
 * $\mathcal R \left({T_1 \cap T_2}\right) = \mathcal R \left({T_1}\right) \cap \mathcal R \left({T_2}\right)$

and:
 * $\displaystyle \mathcal R \left({\bigcap \mathbb T}\right) = \bigcap_{X \in \mathbb T} \mathcal R \left({X}\right)$

where here $\mathcal R = f^{-1}$.

Also see

 * Mapping Image of Intersection


 * Preimage of Intersection


 * Mapping Image of Union
 * Mapping Preimage of Union