proving that the inclusions $i:X \to CX$, $j:Y \to C_f$, $j':Y \to M_f$ have the homotopy extension property

Question

Let $X$ be a topological space, and $f:X \to Y$ a continuous map.
I want to show that the inclusions $i: X \to CX, \: j:Y \to C_f, \: j':Y \to M_f$ all have the homotopy extension property. Where $CX$ denotes the cone over $X$, $C_f$ the mapping cone, and $M_f$ the mapping cylinder.

My ideas:

1. I read that $CX$ has the structure of a relative CW-complex over $X$. Which would already give me the first homotopy-extension property. But I don't understand why this is the case. Since we would have to have uncountably many copies of $X$ to get $X \times [0,1]$

2. I already know that the maps $i,j$ and $j'$ are neighbourhood deformation retracts. But I don't think that this is equivalent in general to HEP.

3. $Y$ should be homotopy-equivalent to $M_f$

Thanks in advance for any help.