Let $M$ be a simplicial model category, $M^o$ its full subcategory of bifibrant objects. The axioms of a simplicial model category guarantee that $M^o$ is enriched in Kan complexes. Thus the homotopy coherent nerve $X = \mathfrak{N}(M^o)$ is a quasicategory, which we think of as the underlying $\infty$-category of $M$.
Let $x,y \in X_0$, then $\text{Map}_X(x,y)$ is the pullback $\text{Fun}(\Delta^1,X) \times_{X \times X} \partial \Delta^1$ of simplicial sets. This is an $\infty$-groupoid i.e. Kan complex. It is well known that if $\underline{M}(x,y)$ denotes the simplicial mapping space in $M$, then there is a weak equivalence $$\underline{M}(x,y) \to \text{Map}_X(x,y)$$ Just by unravelling the definitions it is clear that $\underline{M}(x,y)_0 = \text{Map}_X(x,y)_0$, but how do I prove that the map is a weak equivalence? I have not seen a direct proof of this fact in the literature.
