I'm trying to prove that given a regular continuous time Markov chain $X_t$ (pure jump process), its embedded chain given by $Y_n=X_{T_n}$ is a homogeneous Markov chain, where $T_n$ is the time of the $n$th jump.
This is a classic theorem and is proved in virtually every Markov process book. I've skimmed through quite a lot of them and they usually prove it by using the strong Markov property: if $\tau$ is a finite stopping time, $\forall t>0$ $$P(X_{\tau+t}=y|X_{\tau}=x)=P(X_t=y|X_0=x)$$
Then if $t_n=T_n-T_{n-1}$, by the strong Markov property $$P(X_{T_{n+1}}=y|X_{T_n}, \text{past})=P(X_{t_1}=y|X_0=x)$$
Now, the problem I have is: why is it true that $P(X_{T_n+t_{n+1}}|X_{T_n})=P(X_{t_1}|X_0)$?
I kind of get it intuitively by the strong Markov property, but $t_{n+1}$ is random, I can't just use the strong Markov property there.
Any help is highly appreciated. Regards!