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:
 * $f^{-1} \left[{T_1 \cap T_2}\right] = f^{-1} \left[{T_1}\right] \cap f^{-1} \left[{T_2}\right]$

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 Image of Intersection under One-to-Many Relation applies:
 * $\mathcal R \left[{T_1 \cap T_2}\right] = \mathcal R \left[{T_1}\right] \cap \mathcal R \left[{T_2}\right]$

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

Also see

 * Mapping Image of Intersection


 * Preimage of Intersection


 * Mapping Image of Union
 * Mapping Preimage of Union