Thermoelastic properties of mantle candidate minerals are essential to our understanding of geophysical phenomena, geochemistry, and geodynamic evolutions of the silicate Earth. However, the lower-mantle mineralogy remains much debated due to the lack of single-crystal elastic moduli (C-ij) and aggregate sound velocities of (Al,Fe)-bearing bridgmanite, the most abundant mineral of the planet, at the lower mantle pressure-temperature (P-T) conditions. Here we report single-crystal C-ij of (Al,Fe)-bearing bridgmanite, Mg0.88Fe0.1Al0.14Si0.90O3 (Fe10-Al14-Bgm) with Fe3+/Sigma Fe = similar to 0.65, up to similar to 82 GPa using X-ray diffraction (XRD), Brillouin light scattering (BLS), and impulsive stimulated light scattering (ISLS) measurements in diamond-anvil cells (DACs). Two crystal platelets with orientations of (-0.50, 0.05, -0.86) and (0.65, -0.59, 0.48), that are sensitive to deriving all nine C-ij, are used for compressional and shear wave velocity (v(P) and v(S)) measurements as a function of azimuthal angles over 200 degrees at each experimental pressure. Our results show that all C-ij of singe-crystal Fe10-Al14-Bgm increase monotonically with pressure with small uncertainties of 1-2% (+/- 1 sigma), except C-55 and C-23, which have uncertainties of 3-4%. Using the third-order Eulerian finite-strain equations to model the elasticity data yields the aggregate adiabatic bulk and shear moduli and respective pressure derivatives at the reference pressure of 25 GPa: K-S = 326 +/- 4 GPa, mu = 211 +/- 2 GPa, K-s ' = 3.32 +/- 0.04, and mu ' = 1.66 +/- 0.02 GPa. The high-pressure aggregate v(S) and v(P) of Fe10-Al14-Bgm are 2.6-3.5% and 3.1-4.7% lower than those of MgSiO3 bridgmanite end-member, respectively. These data are used with literature reports on bridgmanite with different Fe and Al contents to quantitatively evaluate pressure and compositional effects on their elastic properties. Comparing with one-dimensional seismic profiles, our modeled velocity profiles of major lower-mantle mineral assemblages at relevant P-T suggest that the lower mantle could likely consist of about 89 vol% (Al,Fe)-bearing bridgmanite. After considering uncertainties, our best-fit model is still indistinguishable from pyrolitic or chondritic models.