Quantifier in front of a relation

Question

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?

Answer 1

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