conditional distribution of $X$ given $\min(X,\alpha)$

Question

let $\alpha \in \mathbb{R}.$ So I want to find a conditional distribution for $X$ given $Y=\min(X,\alpha).$ I know that for all $B \in \mathcal{B}(\mathbb{R})$ $$P_Y(B)=\int_B 1_{]-\infty, \alpha]}(x)dP_X+P(X>\alpha)\delta_{\alpha}(B).$$

I don't know this will help us to find out the conditional distribution. Thank you for any idea.