We consider the fundamental system in electro-magneto-hydrodynamics. This fundamental system was derived by I. Imai in 1962, which consists of the compressible Navier-Stokes equation (5 equations) for fluid part and the Maxwell equation (7 equations with 2 constraints) for electro-magnetic part. To verify the well-posedness of this fundamental system, we first eliminate the electric charge density by using the first constraint. Secondly, we modify the reduced system by using the second constraint. This modified system is equivalent to the original fundamental system and is regarding as a symmetric hyperbolic-parabolic system in the non-relativistic region. Therefore, by applying the general theory, we can prove the time-local well-posedness in the standard $L^2$ type Sobolev space.