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.
If $Af=0$, then, for every $t\geqslant0$, $\mathrm e^{tA}f=f+\displaystyle\sum_{n\geqslant0}\frac{t^{n+1}}{(n+1)!}A^n(Af)=f$. By definition of $A$, $\mathrm e^{tA}=P_t$. Hence $P_tf=f$.