Biyective graph homomorphism implies isomorphism. Given this definition:

If $G_1=(V_1,E_1),G_2=(V_2,E_2)$ are graphs, then $\varphi:V_1\rightarrow V_2$ is a homomorphism iff $\{v_1,v_2\}\in E_1\Rightarrow \{\varphi(v_1),\varphi(v_2)\}\in E_2$

I want to show that if $\varphi$ is a biyective graph homomorphism, then $G_1\cong G_2$.
With this definition of isomorphic graphs:

If $G_1=(V_1,E_1),G_2=(V_2,E_2)$ are graphs, then $G_1\cong G_2$ iff there exist $\phi_1:V_1\rightarrow V_2,\phi_2:E_1\rightarrow E_2$ bijective functions such that:
  $\{v_1,v_2\}\in E_1\Leftrightarrow\phi_2(\{v_1,v_2\})=\{\phi_1(v_1),\phi_1(v_2)\}$.

I defined $\phi_1=\varphi$ and $\phi_2(\{v_1,v_2\})=\{\varphi(v_1),\varphi(v_2)\}$. And I have already proved it's biyective. But I don't know if I got confused, but I haven't been able to prove the $\Leftarrow$ direction of last implication. Any idea?

If $V_1=V_2=\{1,2\}$ and $E_1=\emptyset$ and $E_2=\{\{1,2\}\}$ and $\phi:V_1\to V_2$ is the identity map, then $\phi$ is a bijective homomorphism, but $G_1$ and $G_2$ are not isomorphic.
