Extending a Permutation of [n-1] to a Permutation of [n] with a Prescribed Value, and Its Sign

In the setting of Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices, let $n\in\mathbb{N}$ with $2\le n$, let $m=n-1$, the difference, which lies in $\mathbb{N}$ by The Index Map Skipping One Element of [n] §difference, and let $i\in[n]$, with $[n]$ and $[m]$ as in Intervals of Natural Numbers §segment, the sets $S_{n}$ and $S_{m}$ of permutations of $[n]$ and of $[m]$, whose elements are maps as in Permutations of a Set and the Set of Permutations of [n] §membership, the sign $\operatorname{sgn}$ on $S_{n}$ and on $S_{m}$, for every $p\in[n]$ the deletion index map $s_{p}:[m]\to[n]$, and the powers $(-1)^{k}$ in $\mathbb{R}$, $k\in\mathbb{N}_{0}$, as in Commutative Rings, Fields and Ordered Fields: Standard Notation §rings.

¶ There is exactly one map $\Phi_{i}:[n]\times S_{m}\to S_{n}$ such that, for all $j\in[n]$ and $\tau\in S_{m}$, $\Phi_{i}(j,\tau)(i)=j$ and $\Phi_{i}(j,\tau)(s_{i}(k))=s_{j}(\tau(k))$ for every $k\in[m]$.

¶ The map $\Phi_{i}$ of clause map is a bijection.

¶ For the map $\Phi_{i}$ of clause map and all $j\in[n]$ and $\tau\in S_{m}$, $\operatorname{sgn}(\Phi_{i}(j,\tau))=(-1)^{i+j}\operatorname{sgn}(\tau)$.