Composition of Direct Image Mappings of Relations

Theorem
Let $A, B, C$ be non-empty sets.

Let $\mathcal R_1 \subseteq A \times B, \mathcal R_2 \subseteq B \times C$ be relations.

Let:
 * ${\mathcal R_1}^\to: \mathcal P \left({A}\right) \to \mathcal P \left({B}\right)$

and
 * ${\mathcal R_2}^\to: \mathcal P \left({B}\right) \to \mathcal P \left({C}\right)$

be the mappings induced by $\mathcal R_1$ and $\mathcal R_2$.

Then:
 * $\left({\mathcal R_2 \circ \mathcal R_1}\right)^\to = {\mathcal R_2}^\to \circ {\mathcal R_1}^\to$

Proof
Let $S \subseteq A, S \ne \varnothing$.

Then:

Now we treat the case where $S = \varnothing$: