Suppose we have two random variables $X$ and $Y$, and a region $E$ in the image of $X$. I want to evaluate
$$ P(X \in E | Y = y) \\ = \frac{ P(X \in E \cap Y = y) }{ P(Y=y) } \\ = \frac{ P( Y = y | X \in E ) P(X \in E) }{ P(Y=y) } \\ $$ So far so good. Now to go the next step, I need to sum over values of X. Are the following three all equivalent? Are they all true?: $$ =A = \frac{ \int_{Im X} P( Y = y | X \in E ) P(X \in E | X = x) P(X = x) dx }{ P(Y=y) } \\ $$ or $$ =B = \frac{ \int_{Im X} P( Y = y | X \in E, X = x ) P(X \in E | X = x) P(X = x) dx }{ P(Y=y) } \\ $$
or $$ =C = \frac{ \int_{Im X} P( Y = y | X = x ) P(X \in E | X = x) P(X = x) dx }{ P(Y=y) } \\ $$
How can I tell which of these are equivalent, either mathematically or by visualization/intuition?