I am trying to better understand the ideas and intuition behind the Pollard Rho factorization algorithms.
Given an $x_0$ and an irreducibe polynomial we can create a sequence from the recursive definition:
$x_i \equiv f(x_{i-1}) \mod n$
Let $y_i = x_i \mod d$
The book I am reading then states: Since
$x_i \equiv f(x_{i-1}) \mod n$ then
$y_i \equiv f(y_{i-1}) \mod d$
Why does the change in modulus preserve the relationship $x_i \equiv f(x_{i-1})$?
I am starting to see that since $y_i$ is $x_i$ then the same function with $f(y_{i-1})$ so it is just the same numbers with a different modulus. Is this correct? Is there a better or more formal way to see this?