Preimage of Union under Relation

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

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

Then $$\mathcal R^{-1} \left({T_1 \cup T_2}\right) = \mathcal R^{-1} \left({T_1}\right) \cup \mathcal R^{-1} \left({T_2}\right)$$.

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

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

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

Then:
 * $$\mathcal R^{-1} \left({\bigcup \mathbb T}\right) = \bigcup_{X \in \, \mathbb T} \mathcal R^{-1} \left({X}\right)$$

Proof
This follows from Image of Union, and the fact that $$\mathcal R^{-1}$$ is itself a relation, and therefore obeys the same rules.