Show $\sigma$-algebra of cylindrical sets is a certain collection I am trying to prove that $B\in$ the $\sigma$-algebra generated by the cylindrical sets of the form $\{f: [0,1] \to \mathbb{R} | (f(t_1),...,f(t_n))\in A\}$ for some $n \geq 1$ and Borel subset $A\subset \mathbb{R}^n$, iff $\exists t_1,t_2,...\in [0,1]$, and $C\in \mathcal{F}_0$, where $\mathcal{F}_0$ is the $\sigma$-algebra generated by the cylindrical sets of $\mathbb{R}^{\mathbb{N}}$, s/t $B = \{f\in \{f: [0,1]\to \mathbb{R}\} : (f(t_1), f(t_2), ...)\in C \}$.
I feel lost on this one. Any tips?
Edit: I think I can do this with the good sets principle but I'm having trouble proving that $B = \{f\in \{f: [0,1]\to \mathbb{R}\} : (f(t_1), f(t_2), ...)\in C \}$ is a $\sigma$-algebra. I can show that it's closed under complements but not under countable union or intersection.

Let $(B_n)_{n\geqslant 1}$ be such that 
$$B_n=\{f, (f(t_{n,k}))_{k=1}^\infty\in C_n\},$$
where $C_n$ is a Borel subset of $\mathbb R^{\mathbb N}$ and $t_{n,k}\in [0,1]$. 
Let $\tau\colon\mathbb N\to\mathbb N$ be a bijection. Define 
$$C:=\{(x_j)_{j\in\mathbb N}, x_j\in C_k\mbox{ if }\pi_1(\tau^{-1}(j))=k\},$$
that is $x_j\in C_k$ if $j$ is reached by an element of the row $k$.
(maybe it is simpler we we deal with the union of two elements: in this case, the corresponding $C$ is $(x_j)_{j\in\mathbb N}, x_{2j}\in C_1,x_{2j+1}\in C_2\}$)
