The question is related to the question:    https://mathoverflow.net/questions/309538/detecting-weak-equivalences-in-a-simplicial-model-category/309546?noredirect=1#comment771217_309546

Suppose that we have a simplicial model category $M$ and let denote $M^{f}$ the full simplicial subcategory of fibrant objects. Suppose that $R$ is a subcategory of $M^{f}$ such that for any object $m\in M^{f}$ there exists an objects $r\in R$ such that $r$ is (zigzag) equivalent to $m$ i.e. $r$ and $m$ are isomorphic in $Ho(M)$ the homotopy category of $M$. Let $w: a\rightarrow b$ be a morphism in $M$ such that for any object $r\in R$ the induced map of simplicial sets $w^{\ast}:Map_{M}(b,r)\rightarrow Map_{M}(a,r)$ is a weak homotopy equivalence of simplicial sets. 
Can we conclude that $w$ is a weak equivalence in the model category $M$ ?

If $R=M^{f}$ this is true and is proved in Hirschhorn's book.