Brownian motion where the parameter is a function of $t$

Question

i have a question about this exercise:

Let $f\in L^2[0,t]$ $ \forall t\geq0 $, e $\forall t\geq0$ let $a(t)=\int_{0}^{t}f^2(u)du$ and $Y_t=B_{a(t)}$, where B is a continous $(F_t)_{t\geq0}$-brownian motion. Show that $Y_t$ is a $(F_{a(t)})_{t\geq0}$-brownian motion.

I'm wondering what i have to prove. Isn't it obvious because $B$ is a brownian motion?