If $V$ is a partial isometry with initial projection $E$ then is $EVE=V$?
To your question in the end: That is not true in general. For example, if $\xi,\eta$ are orthonormal vectors, then $V=\langle\xi,\cdot\,\rangle\eta$ is a partial isometry with initial projection $E=\langle\xi,\cdot\,\rangle\xi$, but $EVE=0\neq V$.
To prove this claim in your specific situation, you have to use that $F\leq E_1$, which implies $E_1 F=F$. Thus $$ E_1VE_1=E_1 V=E_1FV=FV=V. $$