Expectation of hitting time of a homogenous Markov chain

Question

My question concerns Exercise 1.15 from the book "Markov Chains and Mixing Times" of Levin, Peres and Wilmer:

For a subset $A\subset \Omega$, define $f(x) = E_x(\tau_A)$. Show that

1. $f(x) = 0$ for $x \in A$.
2. $f(x) = 1 + \sum_{y\in\Omega}P(x, y)f(y)$ for $x \not\in A$.
3. f is uniquely determined by 1. and 2.

Here $\tau_A$ is the hitting time of $A$.

Points 1. and 2. were discussed (in a different form) here. My question relates to point 3. I cannot see where the uniqueness comes from.