  In this short note we present an alternative proof of the so-called First
Visit Time Lemma (FVTL), originally presented by Cooper and Frieze in its first
formulation in [21], and then used and refined in a list of papers by Cooper,
Frieze and coauthors. We work in the original setting, considering a growing
sequence of irreducible Markov chains on $n$ states. We assume that the chain
is rapidly mixing and with a stationary measure having no entry which is too
small nor too large. Under these assumptions, the FVTL shows the exponential
decay of the distribution of the hitting time of a given state $x$ -- for the
chain started at stationarity -- up to a small multiplicative correction. While
the proof of the FVTL presented by Cooper and Frieze is based on tools from
complex analysis, and it requires an additional assumption on a generating
function, we present a completely probabilistic proof, relying on the theory of
quasi-stationary distributions and on strong-stationary times arguments. In
addition, under the same set of assumptions, we provide some quantitative
control on the Doob's transform of the chain on the complement of the state
$x$.
