  We discuss the origin of both non-Abelian discrete family symmetry and
Abelian continuous family symmetry, as well as matter parity, from F-theory
SUSY GUTs. We propose a minimal model based on the smallest GUT group $SU(5)$,
together with the non-Abelian family symmetry $D_4$ plus an Abelian family
symmetry, where fluxes are responsible for doublet-triplet splitting, leading
to a realistic low energy spectrum with phenomenologically acceptable quark and
lepton masses and mixing. We show how a $Z_2$ matter parity emerging from
F-theory can suppress proton decay while allowing neutron-antineutron
oscillations, providing a distinctive signature of the set-up.
