Let $\sigma_i$ be the strategy profile for bidder $i$ that indicates how they should bid based on their value. As we know, if there are 2 bidders both with their values $v_1,v_2$ on $U[0,1]$ in a First-Price Auction, they will bid $\frac{v_1}{2}$ and $\frac{v_2}{2}$ respectively, so $\sigma_1(v_1)=\frac{v_1}{2},\sigma_2(v_2)=\frac{v_2}{2}$.
However, if bidder 1's value is still $U[0,1]$ but bidder 2's value is $U[0,2]$, there does not exist a Bayes-Nash Equilibrium. Why is this? Specifically, why is the following not a BNE?
\begin{equation*} \begin{split} \sigma_1(v_1) &= \frac{v_1}{2} \\ \sigma_2(v_2)&= \begin{cases} \frac{v_2}{2} & \mbox{ if }v_2\in[0,1] \\ \frac{1}{2} & \mbox{ if }v_2>1 \end{cases} \end{split} \end{equation*}
Equilibrium says for all bidders $i$ and values $v_i$, $\sigma_i(v_i)$ is the optimal bid. Is there a bidder and value for which this is not the case?