# Quantifier in front of a relation

If $$R(x,z_1,\dots, z_n)\subset \mathbb{N}^{n+1}$$ is a relation, then is the thing $$(\forall y < x) R(y,z_1,\dots, z_n)$$ a relation? I can't see how to interpret it as a set (relations are sets), so it doesn't look to me that it's a relation. What is it then?

• It looks like a statement (that for any $y$ less than $x$ in some sense, $𝑅(𝑦,𝑧_1,\ldots,𝑧_𝑛)$ holds. Nov 6 '19 at 22:28

This is a formula, which defines a relation. Namely the relation $$\{(x,z_1,\dots,z_n)\in N^{n+1}\mid (y,z_1,\dots,z_n)\in R \text{ for all }y