Inverse of Inverse of Bijection/Proof 1

Theorem
Let $f: S \to T$ be a bijection.

Then:
 * $\left({f^{-1}}\right)^{-1} = f$

where $f^{-1}$ is the inverse of $f$.

Proof
Let $f: S \to T$ be a bijection.

From Bijection Composite with Inverse we have:

where $I_S$ and $I_T$ are the identity mappings on $S$ and $T$ respectively.
 * $f^{-1} \circ f = I_S$, where $I_S$ is the identity mapping on $S$
 * $f \circ f^{-1} = I_T$, where $I_T$ is the identity mapping on $T$

The result follows from Left and Right Inverses of Mapping are Inverse Mapping.