Injective burrows wheeler
Let $\mathsf{BWT}^{-1}$ be the inverse Burrows-Wheeler transform, that is, $\mathsf{BWT}^{-1}(\mathsf{BWT}(w)) = w$.

If $\mathsf{BWT}(x) = \mathsf{BWT}(y)$ then
$$
x = \mathsf{BWT}^{-1}(\mathsf{BWT}(x)) = \mathsf{BWT}^{-1}(\mathsf{BWT}(y)) = y.
$$