  Taylor-Wiles type lifting theorems allow one to deduce that for $\rho$ a
"sufficiently nice" $l$-adic representation of the absolute Galois group of a
number field whose semi-simplified reduction modulo $l$, denoted
$\overline{\rho}$, comes from an automorphic representation then so does
$\rho$. The recent lifting theorems of Barnet-Lamb-Gee-Geraghty-Taylor impose a
technical condition, called \emph{$m$-big}, upon the residual representation
$\overline{\rho}$. Snowden-Wiles proved that for a sufficiently irreducible
compatible system of Galois representations, the residual images are \emph{big}
at a set of places of Dirichlet density $1$. We demonstrate the analogous
result in the \emph{$m$-big} setting using a mild generalization of their
argument.
