Let $X$ be a set. A semi-metric $d$ is a function $d:X\times X\rightarrow \mathbb{R}^+\cup \{\infty\}$ such that a) $d(x,y)>0 \text{ if } x\neq y$, b) $d(x,y)=d(y,x)$, c) $d(x,y)\leq d(y,z)+d(x,z)$, where $x,y,z\in X$. The $d(x,y)=0\implies x=y$ condition of a metric is relaxed.
The following is Proposition 1.1.5 from Metric Geometry by Burago et al. $x\sim y$ iff $d(x,y)=0$. Since $d(x,y)=d(x_1,y_1)$ whenever $x\sim x_1$ and $y\sim y_1$, the projection $\hat{d}$ of $d$ onto $\hat{X}=X/\sim$ is well defined. The $(\hat{X},\hat{d})$ is a metric space.
What is meant by the the projection $\hat{d}$ of $d$ onto $\hat{X}=X/\sim$? Also if it is onto $\hat{X}=X/\sim$, then how is it ensured that the image of $\hat{d}$ is a non-negative real, for it to satisfy the definition of a metric?