Intersection of Injective Image with Relative Complement

Theorem
Let $f: S \to T$ be a mapping.

Then $f$ is an injection :
 * $\forall A \subseteq S: f \sqbrk A \cap f \sqbrk {\relcomp S A} = \O$

Proof
From Intersection with Relative Complement is Empty:
 * $A \cap \relcomp S A = \O$

From Image of Intersection under Injection:
 * $\forall A, B \subseteq S: f \sqbrk {A \cap B} = f \sqbrk A \cap f \sqbrk B$

$f$ is an injection.

Hence the result.