Composition of Inverse Image Mappings of Mappings

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

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

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

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

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

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

Proof
Let $T \subseteq C$.

We have:

It remains to be shown that:


 * $\Img f \cap \map {g^\gets} T = \O \iff \Img {g \circ f} \cap T = \O$

So: