I am trying to compute the transition probabilities of the model given by $X_{n+1} = f_{n+1}(X_n,W_{n+1})$ where $X_n's, W_n's$ are $R^k$ valued random variables for $n \geq 0$, $W_n's$ are independent and $f_n's$ are measurable. Also, define $\mathcal{F}_n=\sigma(X_0,W_1,\cdots,W_n)$. (From this definition we get that $X_n$ is $\mathcal{F}_n$ measurable).
From these notes, the transition probabilites are defined as
$p_{n+1}(X_n,B) := P[X_{n+1}\in B|\mathcal{F}_n]$
Is it true that $ P[X_{n+1}\in B|\mathcal{F}_n] (\omega) = P[f(X_n(w),W_{n+1})\in B]~a.s. $
If so, what is methodology for the proof and if not, what are the transition probabilites for this model?
Thanks for the help.