I am trying to understand why the following is true: $$ p(f(Y) = f(y) \mid Y = y) = p(f(Y) = f(y) \mid X = x, Y = y) \qquad \ldots \text{(Eq. 1)} $$ where $Y$ and $X$ are random variables, and $f(Y)$ is a function of a random variable. $p(Y = y)$ represents the probability density function of random variable $Y$, evaluated at the point $y$.
Equation 1 states the event $f(Y) = f(y)$ conditioned on the value of the random variable $Y$, has no dependence, or is unaffected by the value of random variable $X$. Is there a formal name for Equation 1, or any additional information to help me understand further the process behind Equation 1?
The context for my question is Markov Chains in Information Theory. $X$, $Y$, and $f(Y)$ form a Markov Chain in That Order. Equation 1 must be true to imply the Markov Chain of the three random variables.
Thanks.
$f(Y)$ depends on $Y$, and $X$ depends on $Y$, and $f(Y)$ is conditionally independent of $X$ given $Y$ due to your Markov Chain assumption. Therefore, $p(f(Y)|X,Y) = p(f(Y)|Y)$. In general this is not true, rather $p(f(Y)|X,Y) = \int p(f(Y)|X,Y) p(X|Y) \mathrm{d}X$