How do you formulate the proof of "conditional probability density function $f_{X|Y}(x|y):=\frac{f(x,y)}{f_Y(y)}$ ($f(x,y)$ is the joint probability density function, and $y$ is restricted to the domain where $f_Y(y) \neq 0$ is a probability density function)"?
You can use the properties $\int_{\mathbb R} f_Y(y)dy =1$ and $\int\int_{\mathbb R} f(x,y) dxdy=1$. This seems to be relevant. How do I go on?
Specifically, $\int^{x_1(y)}_{x_2(y)} f_{X|Y}(x|y) = 1$ if the support of this function, i.e. the region where it is non-zero, is $x\in [x_2(y), x_1(y)]$. In these computations, the result being equal to $1$ for all $y$ looks very curious to me.