It is often useful to replace a function with a sequence of smooth functions approximating the given function to resolve minimizing optimization problems. The most famous one is the Moreau envelope. Recently the function was organized using the Bregman distance h D . It is worth noting that Bregman distance h D is not a distance in the usual sense of the term. In general, it is not symmetric and it does not satisfy the triangle inequality The purpose of the research is to study the convergence of the Moreau envelope function and the related proximal mapping depends on Bregman Distance for a function on Banach space. Proved equivalence between Mosco-epi-convergence of sequence functions and pointwise convergence of Moreau-Bregman envelope We also studied the strong and weak convergence of resolvent operators According to the concept of Bregman distance.