Preimage of Composite Relation

Theorem
Let $\mathcal R_1 \subseteq S_1 \times T_1$ and $\mathcal R_2 \subseteq S_2 \times T_2$ be relations.

Let $\mathcal R_2 \circ \mathcal R_1 \subseteq S_1 \times T_2$ be the composition of $\mathcal R_1$ and $\mathcal R_2$.

Then the preimage of $\mathcal R_2 \circ \mathcal R_1$ is given by:


 * $\operatorname{Im}^{-1} \left({\mathcal R_2 \circ \mathcal R_1}\right) = \operatorname{Im}^{-1} \left({\operatorname{Im} \left({\mathcal R_1}\right) \cap \operatorname{Im}^{-1} \left({\mathcal R_2}\right)}\right)$