Permutations of [n] are Maps [n] → [n], and Values of the Determinant are Real Numbers

In the setting of The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness, let $n\in\mathbb{N}$ be a natural number, with $[n]$ as in Intervals of Natural Numbers §segment, the set $S_{n}$ of permutations of $[n]$ as in Permutations of a Set and the Set of Permutations of [n] §permutations, the set $\mathcal{M}_{n}(\mathbb{R})$ of square real matrices, and the determinant $\det$; and maps are as defined there.

¶ For every $\sigma\in S_{n}$, $\sigma$ is a set and $\sigma$ is a map $\sigma:[n]\to[n]$.

¶ For every $A\in\mathcal{M}_{n}(\mathbb{R})$, $\det A\in\mathbb{R}$.