Recently, the mathematical formalism of the Dyson boson mapping has been extended to a system of 3n fermions, leading to the boson-fermion and the baryon mapping. In the present paper, the case of restriction to a subset of three-fermion quantum numbers, the collective indices, is discussed. A theory is developed for the representation of fermionic states and operators in a truncated ideal space where only collective boson-fermion pairs or collective ideal baryons are allowed. An exact reproduction of physical properties is proved to be possible provided that the original fermionic problem can be solved in a subspace where all three-fermion subsystems carry collective indices. Examples of simple applications are presented in the subsequent papers of this series.