It is often proved in Books that the space of Probability measures $\mathcal{P}(S)$ on a Polish metric space $(S,\rho)$ endowed with the weak/narrow topology induced by declaring it to be be the coarsest topology on $\mathcal{P}(S)$, which makes the mappings $$\mathcal{P}(S) \ni \mu \mapsto \int f d\mu \in \mathbb{R}$$ continuous for each bounded and continuous $f : S \rightarrow \mathbb{R}$, is metrizable. Two such metrics should be the Prokhorov metric $d_P$ and the Wasserstein metric $W_0$ of the bounded distance function $\min\{\rho,1\}$.

**But the thing I do not understand is the following:** It is often shown (For example in Villani, 2009, Optimal Transport) that $(\mu_n) \subset \mathcal{P}(S)$ converging weakly to some $\mu \in \mathcal{P}(S)$, that is,
$$\int f d\mu_n \rightarrow \int f d\mu$$
for each bounded and continuous $f : S \rightarrow \mathbb{R}$, is equivalent to $W_0(\mu_n,\mu) \rightarrow 0$, or $d_P(\mu_n,\mu)\rightarrow 0$. If we do not know a priori that the weak topology is metrizable, then we cannot conclude by the above, that the topology generated by $W_0$ or $d_P$ is exactly the weak topology. Or am I missing something?