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 $f \left({a}\right) = f \left({b}\right) \implies a = b$.

So suppose $f \left({a}\right) = f \left({b}\right)$.

Then:

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

So we have shown that $f \left({a}\right) = f \left({b}\right) \implies a = b$.

Hence the result from the definition of injection.

Also see

 * Surjection if Composite is a Surjection