The emergence of the concept of a causal fermion system is revisited and further investigated for the vacuum Dirac equation in Minkowski space. After a brief recap of the Dirac equation and its solution space, in order to allow for the effects of a possibly nonstandard structure of spacetime at the Planck scale, a regularization by a smooth cutoff in momentum space is introduced, and its properties are discussed. Given an ensemble of solutions, we recall the construction of a local correlation function, which realizes spacetime in terms of operators. It is shown in various situations that the local correlation function maps spacetime points to operators of maximal rank and that it is closed and homeomorphic onto its image. It is inferred that the corresponding causal fermion systems are regular and have a smooth manifold structure. The cases considered include a Dirac sea vacuum and systems involving a finite number of particles and antiparticles.