$\{Y_i, \forall i \ge 0\}$ is iid random variable, is this equation true?

Question

$\{Y_i, \forall i \ge 0\}$ is iid random variable, is this equation true
$$ p(g(Y_{n+1}) = k_{n+1} | f_{n}(Y_0,Y_1,...,Y_n) = k_n, f_{n-1}(Y_0,Y_1,...,Y_{n-1}) = k_{n-1},...,f_0(Y_0) = k_0) = p(g(Y_{n+1}) = k_{n+1}) $$
I know $g(Y_{n+1})$ is independent of every $f_i(Y_0, ...,Y_i), \forall i \le n$

Answer 1

Hint: If $A$ is independent of $B$, then $$p(A\mid B) \equiv \frac{p(A \cap B)}{p(B)} = \frac{p(A) p(B)}{p(B)} = p(A).$$