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Given a canonical Feller process $(X_t,P_x)$ with Feller semigroup $P$. Let $T$ a (good) stopping time, for example $T=\inf\{u\ge 0 : X_u=0\}$. I'm looking for a proof of the following claim $$\hat{P}_t f(x)= E_x[f(X_t) 1_{\{t<T\}}]$$ is a Feller semigroup. I looked in the classical books Blumenthal, Walsh, Sharpe but without finding an answer.

I would appreciate any help from you.

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