Determinant of Kronecker Delta Elements

Theorem
Let $\lambda$ and $\pi$ be permutations on $\set {1, 2, 3}$.

Let:
 * $\tuple {i, j, k} = \tuple {\map \lambda 1, \map \lambda 2, \map \lambda 3}$


 * $\tuple {r, s, t} = \tuple {\map \pi 1, \map \pi 2, \map \pi 3}$

Then:
 * $\begin {vmatrix}

\delta_{i r} & \delta_{i s} & \delta_{i t} \\ \delta_{j r} & \delta_{j s} & \delta_{j t} \\ \delta_{k r} & \delta_{k s} & \delta_{k t} \end {vmatrix} = \map \sgn {i, j, k} \map \sgn {r, s, t}$

where:
 * $\delta_{ir}$ denotes the Kronecker delta
 * $\begin{vmatrix} \cdot \end{vmatrix}$ denotes a determinant
 * $\map \sgn {i, j, k}$ is the sign of the permutation $\tuple {i, j, k}$ of the set $\set {1, 2, 3}$.