We study the condensed phase of a Bose-Fermi mixture with a tunable pairing interaction between bosons and fermions with many-body diagrammatic methods and fixed-node diffusion quantum Monte Carlo simulations. A universal behavior of the condensate fraction and bosonic momentum distribution with respect to the boson concentration is found to hold in an extended range of boson-fermion couplings and concentrations. For vanishing boson density, we prove that the bosonic condensate fraction reduces to the quasiparticle weight Z of the Fermi polaron studied in the context of polarized Fermi gases, unifying in this way two apparently unrelated quantities. Bose-Fermi (BF) mixtures with a tunable pairing interaction between bosons and fermions have been actively investigated in the context of ultracold gases , where the tunability of the BF interaction has been demonstrated and exploited in several experiments [23][24][25][26][27][28][29][30][31][32]. Previous work has shown that, even at zero temperature, a sufficiently strong BF attraction suppresses completely the boson condensate in mixtures where the number of bosons does not exceed the number of fermions [1,12,15]. This is due to pairing of bosons with fermions into molecules, which competes with condensation in momentum space. In particular, a first-order phase transition from a superfluid phase with a bosonic condensate, to a normal (molecular) phase without a condensate was recently demonstrated with fixed-node diffusion Monte Carlo (FNDMC) simulations [19].Here, we focus on the superfluid phase at zero temperature and present a many-body diagrammatic formalism able to describe this phase from weak to strong BF coupling. Our approach is validated by comparing it with previous [19] and new dedicated FNDMC calculations. By using both methods, we then analyze the condensate fraction and the momentum distributions, and establish a remarkable connection with the polaron problem in polarized Fermi gases.Model and diagrammatic formalism. The system of our interest is a mixture of bosons of mass m B and number density n B , interacting with spinless fermions of mass m F and number density n F . The system is dilute, such that the range of all interactions can be considered smaller than the relevant interparticle distances. The BF pairing interaction can be described then by an attractive contact potential, whose strength is parametrized in terms of the BF scattering length a BF with the same regularization procedure commonly used for Fermi gases [33,34]. The interaction between bosons is instead assumed to be repulsive, with scattering length a BB of the order of the interaction range. No interaction between fermions is considered, since short-range interactions are suppressed by Pauli principle. We are interested in systems with concentration of bosons x = n B /n F 1, where a full competition between pairing and condensation is allowed. A natural (inverse) length scale is then provided by the Fermi wave vector k F ≡ (6π 2 n F ) 1/3 , which can be combined with a BF to...