Let $A \subset \mathbb{R}^{n}$ be open in $\mathbb{R}^{n}$ and let $F: A \to \mathbb{R}^{n} \times \mathbb{R}^{n}$ be continuous. Then $F$ is called a tangent vector field on $A$ if and only if $F(x) \in \{x\} \times \mathbb{R}^{n}$ for all $x \in A$.
I am trying to reduce this definition to see if it is possible to somehow write the part "$F(x) \in \{ x \} \times \mathbb{R}^{n}$" into the part "$F: A \to \mathbb{R}^{n} \times \mathbb{R}^{n}$". In other words, I am working to see if the definition can be written as the usual functional notation such as $f: x \mapsto x^{2} : \mathbb{R} \to \mathbb{R}$, which is more appealing. But so far I have come up with no simpler methods.