  In this paper, we study the relativistic energy spectrum for Dirac fermions
under rainbow gravity effects in the $(3+1)$-dimensional Bonnor-Melvin-Lambda
spacetime, where we work with the curved Dirac equation in cylindrical
coordinates. Using the tetrads formalism of General Relativity and considering
a first-order approximation for the trigonometric functions, we obtain a Bessel
equation. To solve this differential equation, we also consider a region where
a hard-wall confining potential is present (i.e., some finite distance where
the radial wave function is null). In other words, we define a second boundary
condition (Dirichlet boundary condition) to achieve the quantization of the
energy. Consequently, we obtain the spectrum for a fermion/antifermion, which
is quantized in terms of quantum numbers $n$, $m_j$ and $m_s$, where $n$ is the
radial quantum number, $m_j$ is the total magnetic quantum number, $m_s$ is the
spin magnetic quantum number, and explicitly depends on the rainbow functions
$F(\xi)$ and $G(\xi)$, curvature parameter $\alpha$, cosmological constant
$\Lambda$, fixed radius $r_0$, and on the rest energy $m_0$, and $z$-momentum
$p_z$. So, analyzing this spectrum according to the values of $m_j$ and $m_s$,
we see that for $m_j>0$ with $m_s=-1/2$ (positive angular momentum and spin
down), and for $m_j<0$ with $m_s=+1/2$ (negative angular momentum and spin up),
the spectrum is the same. Besides, we graphically analyze in detail the
behavior of the spectrum for the three scenarios of rainbow gravity as a
function of $\Lambda$, $r_0$, and $\alpha$ for three different values of $n$
(ground state and the first two excited states).
