Given some probability space, say $(\Omega, \mathcal{F}, \mathbb{P})$, and the fact that $\mathcal{X}$ is $\mathbb{F}$-adapted, and $\mathcal{X}$ is Markov, then I know the following is true:
$\mathbb{P}(\mathcal{X_t} \in A | \mathcal{F}_s) = \mathbb{P}(X_t \in A | \mathcal{X}_s)$
...which follows directly from the fact that $\mathcal{X}_t - \mathcal{X}_s \mathrel{\unicode{x2AEB}} \mathcal{X}_s \, \forall s \le t$
But is it also true to say these two claims:
1) $\mathcal{X}_t \mathrel{\unicode{x2AEB}} \mathcal{F}_s | \mathcal{X}_s \, \forall s \le t$
2) $\mathbb{E}\mathcal{X}_t \mathrel{\unicode{x2AEB}} \mathcal{F}_s | \mathcal{X}_s \, \forall s \le t$
It seems to be true intuitively but I cannot derive mathematically.