Normality is preserved under any monotone subgroup-defining function

Statement
Suppose $$f$$ is a monotone subgroup-defining function, i.e., $$f$$ is a subgroup-defining function such that whenever $$H \le G$$ are groups, $$f(H) \le f(G)$$. Then, the following is true:

If $$H \le G$$ and $$H$$ is a normal subgroup of $$G$$, then $$f(H)$$ is a normal subgroup of $$f(G)$$.

Facts used

 * 1) uses::Subgroup-defining function value is characteristic
 * 2) uses::Characteristic of normal implies normal
 * 3) uses::Normality satisfies intermediate subgroup condition

Proof
Given: $$H$$ is a normal subgroup of $$G$$, $$f$$ is a monotone subgroup-defining function.

To prove: $$f(H)$$ is a normal subgroup of $$f(G)$$.

Proof: