Let $X$ be a RCLL Markov Process with generator $A$. Then I know that
$ M^f = f(X)-f(X_0)-\int Af(X_s)ds $
is a martingal for every $f\in \mathcal{D}_A$. If we suppose that $Af=0$, we see that $f(X)-f(X_0)$ is a martingale. Further I know that from the Markov property
$E[f(X_{t+h})|\mathcal{F}_t]=P_hf(X_t)$
Where $(P_t)$ is the transition semigroup. Let $X_0=x$ be the starting point. If we assume that $f(X)-f(x)$ is a martingale, then
$P_tf(x)-f(x)=E[f(X_t)-f(x)]=0$
Why is this equation true? I wanted to use the property above with conditional distribution, without success.