**Before I ask my question let me clarify some notation:**
$f^{i,j}_r$, where $i < j$, refers to the inclusion map $f: H_r(X_i) \hookrightarrow H_r(X_j)$. $X_i$ and $X_j$ are subcomplexes of a filtered complex $X$, where $X_i \subseteq X_j$.

I need clarification/explanation of the following definition. I was able to understand birth events easily, but I'm having some trouble with death events. Why do the two conditions below imply the end of a certain feature?

Suppose an $r$-class $\alpha$ is born at the $i$th level. Then $\alpha$ is said to die at level $j$ iff:

- $f^{i,j}_r(\alpha) \in \operatorname{im}(f^{i-1, j}_r)$
- $f^{i,j-1}_r(\alpha) \notin \operatorname{im}(f^{i-1, j-1}_r)$

A death event then refers to the death of all homology classes in the coset $\alpha + \operatorname{im}(f^{i-1,i}_r)$.

**The bounty is for whoever answers my question in the comments below.**