If I have a joint pdf of the following form $$ f(X \leq x, Y = y) $$ where $X$ and $Y$ are continuous random variables. Can the definition of conditional probability be applied such that $$ f(X \leq x, Y = y) = P(X \leq x|Y=y)f(Y=y) $$ ?
This seems valid to me, but is $P(X \leq x|Y=y)$ a probability or a pdf in itself?
It is a conditional cumulative distribution function; so measures a probability mass.