Injective iff it has a left inverse

Statement

Let $A$ be a nonempty set, let $B$ be a set and let $f \colon A \to B$ be a function. Then $f$ is injective if and only if $f$ has a left inverse.

Remarks

The hypothesis that $A$ is nonempty is necessary. The empty function from $\varnothing$ to a nonempty set is injective, but there is no function from a nonempty set to $\varnothing$. To build a left inverse, fix $a_0 \in A$. Send each $b = f(a)$ to $a$, and send each $b \notin f(A)$ to $a_0$.

Explanations

L1 human

An injective function loses no information, so you can go back from each value to its unique input. The values that $f$ never takes can go anywhere, and you need one point $a_0$ to send them to.