Inverse of Permutation is Permutation

Theorem
If $f$ is a permutation of $S$, then so is its inverse $f^{-1}$.

Proof
Let $f: S \to S$ is a permutation of $S$.

By definition, a permutation is a bijection such that the domain and codomain are the same set.

From Bijection iff Inverse is Bijection, it follows $f^{-1}$ is a bijection.

From the definition of inverse relation, the domain of a relation is the codomain of its inverse and vice versa.

Thus the domain and codomain of $f^{-1}$ are both $S$ and it follows that $f^{-1}$ is a permutation.