Let $(X_{t})_{t \in \mathbb R_{+}}$ be some stochastic process defined on the filtered probability space $(\Omega,\mathcal{F}, (\mathcal{F}_{t})_{t\in\mathbb R_{+}},\mathbb P)$ with values in $(\mathbb R, \mathcal{B}(\mathbb R))$ and then define the sigma-algebra $\mathcal{P}_{r}=\{ A\subseteq \mathbb R_{+}\times \Omega: (t,\omega)\mapsto 1_{A}(t,\omega) \text{ is progressively measurable}\}$. Show:
$$ X \text{ is progressively measurable }\iff X \text{ measurable with respect to }\mathcal{P}_{r}$$
My attempt:
$"\Leftarrow"$ For $t > 0$ consider the map $X^{t}: ([0,t]\times \Omega,\mathcal{B}([0,t])\otimes \mathcal{F}_{t})\to (\mathbb R,\mathcal{B}(\mathbb R)), \; (s,\omega)\mapsto X_{s}(\omega)$
Let $B\in \mathcal{B}(\mathbb R)$, we want to show that $X^{t}$ is $\mathcal{B}([0,t])\otimes \mathcal{F}_{t}-$measurable:
$\{X^{t}\in B\}=\{X\in B, s \in [0,t]\}=(A\cap [0,t])\times \Omega$ with $A \in \mathbb R_{+}$ such that $1_{A}^{t}:[0,t]\times\Omega \to (\mathbb R,\mathcal{B}(\mathbb R))$ is $\mathcal{B}([0,t])\otimes \mathcal{F}_{t}$-measurable and thus $\{1_{A}^{t}=1\}=A\times \Omega \in \mathcal{B}([0,t])\otimes\mathcal{F_{t}}\implies A\in \mathcal{B}([0,t])\implies \{X^{t}\in B\}=(A\cap [0,t])\times \Omega\in \mathcal{B}([0,t])\otimes\mathcal{F_{t}}$.
"$\Rightarrow$" Let $B \in \mathcal{B}(\mathbb R)$ and consider the set:
$\{ X \in B\}=\{ (s,\omega)\in \mathbb R_{+}\times \Omega: X_{s}(\omega)\in B\}=\bigcup\limits_{t\in \mathbb R_{+}}\{ (s,\omega)\in [0,t]\times \Omega: X_{s}(\omega)\in B\}$
By assumption we know that $\{ (s,\omega)\in [0,t]\times \Omega: X_{s}(\omega)\in B\}\in \mathcal{B}([0,t])\otimes \mathcal{F}_{t}$ and I do not know how to proceed:
Is the $\Leftarrow$ direction correct? Any tips on the $\Rightarrow$?