Conditional Probability vs Conditional Probability wrt a RV I have some problems understanding how the following conditional probabilities relate.
Given X and Y two continuous rv, we can write the conditional probability of $X\leq x$ given Y as:
\begin{equation}
P[X\leq x|Y]= E[\mathbb{1}_{X\leq x}|Y]
\end{equation}

For the same two variables we can write the following conditional probability distribution:

\begin{equation}
P[X\leq x|Y\leq y]= \frac{P[X\leq x,Y\leq y]}{P[Y\leq y]}=\frac{E[\mathbb{1}_{X\leq x}\mathbb{1}_{Y\leq y}]}{E[\mathbb{1}_{Y\leq y}]}
\end{equation}
Question: What is the relation between the two definitions above and if possible how can we get from one to the other?
Thank you.
$P[X\leq x|Y]= E[\mathbb{1}_{X\leq x}|Y]$ is in fact for a constant $Y=y$, it would be more clear to write it as: 

$P[X\leq x|Y]= E[\mathbb{1}_{X\leq x}|Y=y]$ 

Note that here you are taking the conditional expectaion when you fix $Y$.
But the second formula is just the application of Bayes rule to the first definition. In fact this gives you the intuition on how you can marginalize the joint distribution in a more flexible way.
$P[X\leq x|Y]= E[\mathbb{1}_{X\leq x}|Y]$ is in fact for a constant $Y=y$, it would be more clear to write it as: 

$P[X\leq x|Y]= E[\mathbb{1}_{X\leq x}|Y=y]$ 

Note that here you are taking the conditional expectaion when you fix $Y$.
But the second formula is just the application of Bayes rule to the first definition. In fact this gives you the intuition on how you can marginalize the joint distribution in a more flexible way.