Complement of Direct Image Mapping of Injection equals Direct Image of Complement

Theorem
Let $f: S \to T$ be an injection.

Let $f^\to: \powerset S \to \powerset T$ denote the direct image mapping of $f$.

Then:
 * $\forall A \in \powerset S: \map {\paren {\complement_{\Img f} \circ f^\to} } A = \map {\paren {f^\to \circ \complement_S} } A$

where $\circ$ denotes composition of mappings.

Proof
As $f$ is an injection, it is a fortiori a one-to-many relation.

From Image of Set Difference under Relation: Corollary 2 we have:
 * $\forall A \in \powerset S: \map {\paren {\complement_{\Img {\mathcal R} } \circ \mathcal R^\to} } A = \map {\paren {\mathcal R^\to \circ \complement_S} } A$

where $\mathcal R \subseteq S \times T$ is a one-to-many relation on $S \times T$.

Substituting $f$ for $\mathcal R$ gives the result:
 * $\forall A \in \powerset S: \map {\paren {\complement_{\Img f} \circ f^\to} } A = \map {\paren {f^\to \circ \complement_S} } A$