  We show how the phases of interacting particles in topological flat bands,
known as fractional Chern insulators, can be adiabatically connected to
incompressible fractional quantum Hall liquids in the lowest Landau-level of an
externally applied magnetic field. Unlike previous evidence suggesting the
similarity of these systems, our approach enables a formal proof of the
equality of their topological orders, and furthermore this proof robustly
extends to the thermodynamic limit. We achieve this result using the hybrid
Wannier orbital basis proposed by Qi [Phys. Rev. Lett. 107, 126803 (2011)] in
order to construct interpolation Hamiltonians that provide continuous
deformations between the two models. We illustrate the validity of our approach
for the groundstate of bosons in the half filled Chern band of the Haldane
model, showing that it is adiabatically connected to the $\nu=1/2$ Laughlin
state of bosons in the continuum fractional quantum Hall problem.
