About the metrizability of the space of Probability measures $\mathcal{P}(S)$

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?

This is quite standard, though. The key point is to realize that it is sufficient to check the continuity of $$\mu\mapsto\int f d\mu$$ for a countable number of continuous and bounded functions. This is quite obvious if the space is compact (because in this case $$C_b(S)$$ is separable), but possible also for general Polish spaces.