  We investigate the topological aspect of the spin-triplet $f$-wave
superconductor UPt$_3$ through microscopic calculations of edge- and
vortex-bound states based on the quasiclassical Eilenberger and Bogoliubov-de
Gennes theories. It is shown that a gapless and linear dispersion exists at the
edge of the $ab$-plane. This forms a Majorana valley, protected by the mirror
chiral symmetry. We also demonstrate that, with increasing magnetic field,
vortex-bound quasiparticles undergo a topological phase transition from
topologically trivial states in the double-core vortex to zero-energy states in
the normal-core vortex. As long as the $d$-vector is locked into the
$ab$-plane, the mirror symmetry holds the Majorana property of the zero-energy
states, and thus UPt$_3$ preserves topological crystalline superconductivity
that is robust against the crystal field and spin-orbit interaction.
