# How to show that $\rho(x,y) := \inf\limits_{f\in \mathcal{F}}\{f(x,y)\}$ defines a metric on $X$?

Let $$(X,d)$$ be a metric space and $$\mathcal{F}$$ a family of functions $$f : X \times X \rightarrow [0,\infty)$$ s.t.

(i) $$f(x,y) \geq d(x,y) \ \forall f \in \mathcal{F}$$

(ii) for each $$x \in X$$ there exists $$f \in \mathcal{F}$$ s.t. $$f(x,x) = 0$$

(iii) for each $$f \in \mathcal{F}$$ there exists $$g \in \mathcal{F}$$ s.t. $$g(x,y) = f(y,x)$$ for all $$x,y \in X$$

(iv) for any point $$a \in X$$ and $$f,g \in \mathcal{F}$$ we have that the function $$h : X\times X \rightarrow \mathbb{R}$$ def. by $$h(x,y) = f(x,a) + g(y,a)$$ is in $$\mathcal{F}$$

Show that $$\rho(x,y) := \inf\limits_{f\in \mathcal{F}}\{f(x,y)\}$$ defines a metric on $$X$$

I'm stuck on trying to prove triangle inequality $$\rho(x,y) \leq \rho(x,a) + \rho(a,y)$$. Haven't really got anywhere useful. Any ideas?

Fix $$f, g \in \mathcal{F}$$ and $$a \in X$$. Let $$h$$ be as in condition (iv). Then $$f(x, a) + g(y, a) = h(x, y) \ge \rho(x, y).$$ Since this holds for any such $$f$$, (leaving $$g, a, x, y$$ fixed), then taking the infimum of $$f$$ over $$\mathcal{F}$$ implies $$\rho(x, a) + g(y, a) \ge \rho(x, y).$$ Similarly, leaving $$a, x, y$$ fixed, and taking the infimum over $$g$$, $$\rho(x, a) + \rho(y, a) \ge \rho(x, y).$$ This holds over all $$a, x, y \in X$$, i.e. we've proven the triangle inequality.
Let $$\varepsilon >0$$ and $$f,g$$ s.t. for all $$x,y$$, $$f(x,a)+g(a,y)\leq \rho(x,a)+\rho(a,y)+\varepsilon$$
Since $$h(x,y):=f(x,a)+g(a,y)\in \mathcal F,$$ we get $$\rho(x,y)\leq h(x,y)\leq \rho(x,a)+\rho(a,y)+\varepsilon ,$$ what prove the claim.