When do dependent random variables passed through nonlinear functions have the same conditional PDF?

Question

Say there are 3 random variables $x$ and $y$ where $y$ depends on $x$ through $$y=f(x).$$ Furthermore, say there is another random variable $z$ that depends on $y$ through $$z=g(y).$$ Then, when is the relation $$ p(x\mid y) = p(x\mid z),$$ valid ($p$ is the pdf of a random variable)? Is it only when $g$ and $f$ are both invertible?

Answer 1

For example, when $f$ is not invertible, but $g(y)=y$, you certainly have your relation, so invertibility on both is not required.