Transfinite Recursion Theorem/Uniqueness

Theorem
Let $f$ be a mapping 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 mapping 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, $\forall x \in y: f(x) = g(x)$.

Proof
We shall prove this by induction. Suppose that $\forall x \in \alpha: f(x) = g(x)$ for some arbitrary ordinal $\alpha < y$. Then $\alpha < z$.

So, applying induction, $\forall \alpha < y: f(\alpha) = g(\alpha)$