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.