In computer science I'm used to using overloaded operators. Is this also valid in mathematical notation? Concretely, I have the following example:
Definition:
Let $G_1=(V_1,E_1)$ and $G_2=(V_2,E_2)$ be two graphs. We define $\phi = \{(v,w) \in V_1 \times V_2 | v \text{ is mapped to } w \}$ be a mapping between vertices from these graphs, called an FU instance.
We can say that a vertex $v_i$ is part of an FU instance $\phi$ iff: $\exists (v_i,x) \in \phi \lor \exists (x,v_i) \in \phi$; short-hand notation: $v \in \phi$.
Usage in question:
Let $G=(V_G,E_G)$ be a graph, and $\Phi_G$ be a set of FU instances. We define:
$V^*_G = \{ v \in V_G | n_v > 1\}$ where $ n_v = \#\{ \phi \in \Phi_G | v \in \phi \}$.
Is it appropriate to overload $\in$ in one statement, like that? In the case of $\phi \in \Phi_G$ it's the canonical set membership relation, while in $v \in \phi$ its my self-defined use.
If not, I can easily define my shorthand using a different symbol, but I feel this would make things harder to parse?
I think it's inappropriate, because $v\in \phi$ already has a meaning. I would define a new object, say $$\phi^*=\{v\in V_1|\exists w\in V_2, (v,w)\in\phi\}\cup\{v\in V_2|\exists w\in V_1,(w,v)\in\phi\}$$ and then write $v\in \phi^*$.