Two definitions of topological mixing

Question

I have seen two definitions of topological mixing, and wanted to check that they are equivalent:

1) We call a continuous map $f:X \rightarrow X$ mixing if for every pair of non-empty open sets, $U,V \subset X$, there is some $n_0 \in \mathbb{N}$ such that for all $n> n_0$, $f^{-n}(U) \cap V \neq \emptyset$.

2) We call a continuous map $f:X \rightarrow X$ mixing if for every pair of non-empty open sets, $U,V \subset X$, there is some $n_1 \in \mathbb{N}$ such that for all $n> n_1$, $f^{n}(U) \cap V \neq \emptyset$.

Writing out the sets, it looks like we just interchange the roles of $U$ and $V$, so the definitions should be equivalent, but I wanted to check I wasn't being silly.

Answer 1

Correct: if $x\in f^{-n}(U)\cap V$ then $f^n(x) \in U\cap f^n(V) $.
For the other direction, if $y\in U\cap f^n(V) $ then there is an $x\in V$ such that $y=f^n(x) $ so $x\in f^{- n}(U) \cap V$.