Composition of Direct Image Mappings of Mappings

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

Let $f: A \to B$ and $g: B \to C$ be mappings.

Let:
 * $f^\to: \powerset A \to \powerset B$

and
 * $g^\to: \powerset B \to \powerset C$

be the direct image mappings of $f$ and $g$.

Then:
 * $\paren {g \circ f}^\to = g^\to \circ f^\to$