Injection if Composite is Injection

Theorem
Let $f$ and $g$ be mappings such that their composite $g \circ f$ is an injection.

Then $f$ is an injection.

Proof
Let $g \circ f$ be injective.

We need to show that $\map f a = \map f b \implies a = b$.

So suppose $\map f a = \map f b$.

Then:

and so $a = b$ as $g \circ f$ is an injection.

So we have shown that:
 * $\map f a = \map f b \implies a = b$

Hence the result from the definition of injection.

Also see

 * Surjection if Composite is Surjection