First-visit probabilities into a closed class — when they're forced to be 1, and when they're not

Suppose $f_{ij}$ denotes the probability of the first visit to $j$ from $i$ in a Markov chain with finite state space. (I) If C is a closed communicating class of persistent states, then for any transient state $i$ which leads to $j,k\in C$, $f_{ij}=f_{ik}$. (II) If C is a single closed communicating class and if $i\to j$ where $i\notin C$ and $j\in C$, then $f_{ij}=1$.