Let $X \sim Poisson(\lambda)$ and $Y \sim Poisson(\lambda p)$, where $p\in(0,1)$. How to prove $Y|X=x$ will follow a binomial distribution with $x$ and $p$ as parameters for $x = 0,1,2,...$ i.e.
$$f_{Y|X}(y|x)= \binom{x}{y}p^y(1-p)^{x-y}$$
I know the answer will be $f_{Y|X}(y|x)$, but I am stuck at the steps of how to prove it.