inverse of an injection

Injections are invertible, i.e. the inverse of injections are functions. Non-injective functions cannot have an inverse function. For example, say $f(a_1)=b$ ad $f(a_2)=b$, the inverse relation must contain the ordered pairs $⟨b,a_1⟩$ and $⟨b,a_2⟩$. As such, the inverse cannot be a function since the same argument $b$ cannot be mapped to two different values.

For an injection $f:A\to B$ and its inverse $f^{-1}:f[A]\to A$, $f(f^{-1}(y))=y$ for any $y\in f[A]$.