Let's say we have random variables $X$ and $Y$, $x,y \in \mathbb{R}^n$ with densities $f_X(x)$, $f_Y(y)$, and we also have $y = G(x)$ (where $G$ is a bijective, differentiable function). I know that $f_{X}(x) = f_Y(y) |\det{(\frac{\partial y}{\partial x})}|$, but what can be said about $f_{X|Y_1}(x|y_1)$?
It seems like the density should be zero where $G(x)_1 \neq y_1$ and infinite where $G(x)_1 = y_1$, so more specifically, can we find $\lim\limits_{\epsilon \to 0} f_{X|Y_1}(x|z-\frac{\epsilon}{2} <y_1<z+\frac{\epsilon}{2}) \times \epsilon$? ($f_{X|Y_1}(x|...)$ is the conditional probability density of $x$ given $y_1$ is in the range $(z-\frac{\epsilon}{2}, z+\frac{\epsilon}{2}))$