Lemma 13.6 $p \in F \Longleftrightarrow [p] \in F'$
Proof. The direction $\Rightarrow$ is immediate from the definition of $F'$. For the direction $\Leftarrow$, we need to show that if $p \approx q$ and $p \in F$, then $q \in F$. In other words, every $\approx$-equivalence class is either a subset of $F$ or disjoint from $F$. This follows immediately by taking $x = \epsilon$ in the definition of $p \approx q$. $\quad\square$
The excerpt above found in Dexter C. Kozen's Automata and Computability. This lemma is found in a proof of the Quotient Construction for DFA.
I do not understand how the author concludes that if one wants to show $[p] \in F' \implies p \in F$ (1), then it is equivalent to showing that $p≈q$ and $p \in F \implies q \in F$ (2). Can someone please provide as detailed as possible an explanation of why if we want to show (1) it is equivalent to show (2)? Any idea on why the author does not feel the need to justify this?