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