Conditional probability and relation between random variables

Question

Let's have three random variables $X,Y,Z\in \mathbb{R}$, $X$ and $Y$ are independent and let's assume that $\forall A \in B(\mathbb{R}), \forall y \in \mathbb{R}$, $P(Z \in A|Y=y)=P(Xy \in A)\bigr)$. Where $B(\mathbb{R})$ is the Borel $\sigma$-algebra on $\mathbb{R}$.

Now, is it true that $\forall A \in B(\mathbb{R}), P(Z\in A)=P(XY \in A)$?

Can you prove it or can you suggest me how to prove it?