When we have two random variables $X$ and $Y$ with joint density $f_{X,Y}(x,y)$, we can find the CDF by basically integrating on all the values up to $x$ and $y$ for each variable. Now, I'm wondering if this is true for any set, for example, if I'm interested in $\mathbb{P}(X\in A, Y\in B)$, is the following correct provided that $A$ and $B$ are measurable sets?
$$\mathbb{P}_{X,Y}(X\in A, Y\in B)=\int_A\int_B f_{x,y}(x,y)dydx$$
What I'm trying to figure out in particular is if it is always true that
$$f_{x\mid Y}(x\mid Y\in B)=\frac{1}{\mathbb{P}_y(Y\in B)}\int_B f_{x,y}(x,y)dy$$
We have that
$$f_{X\mid Y}(x\mid Y\in B)=\mathbb{E}(\mathbb{I}(X=x)\mid Y\in B)=\frac{\mathbb{E}(\mathbb{I}( X=x)\mathbb{I}(Y\in B))}{\mathbb{P}_{Y}(Y\in B)}=$$ $$=\frac{1}{\mathbb{P}(Y\in B)}\int_B\int_{X=x}f_{X,Y}(x,y)dxdy=\frac{1}{\mathbb{P}(Y\in B)}\int_Bf_{X,Y}(x,y)dy$$