Joint distribution of random variable and its cdf.

Question

Let $Y:=(X,F_X(X))$ where $X\sim F_X$ and $Y\sim F_Y$. Naturally using the probability integral transform I know that the marginal distribution of the second variable in Y is standard uniformly distributed so I define $U:=F_X(X)$. Now, intuitively I would say that $$F_{X|U=F_X(x)}(x) = P(X\leq x|U=F_X(x))=P(X\leq x)=F_X(x),$$ but how would I go about proving it?