Transfinite Recursion Theorem/Uniqueness

Theorem
Let $f$ be a function with a domain $y$ where $y$ is an ordinal.

Let $f$ satisfy the condition that $\forall x \in y: f(x) = G(f\restriction x)$

Let $g$ be a function with a domain $z$ where $z$ is an ordinal.

Let $g$ satisfy the condition that $\forall x \in z: f(x) = G(f\restriction z)$

Let $y \subseteq z$.

Then, $\forall x \in y: f(x) = g(x)$.