How does the boiling point of water inside a pressure cooker change with temperature? [closed]

My question is why a pressure cooker would affect the boiling point at all.

To my understanding, the boiling point of a substance is defined by 'The temperature at which the vapor pressure of the substance equals the surrounding pressure' and according to the Wikipedia article regarding pressure cooking, 'In a sealed pressure cooker, the boiling point of water increases with increasing pressure'.

Now say we fill a pressure cooker with water and turn on the heat for a while. Eventually, it'll start vaporizing and all the pressure accumulated inside the pressure cooker due to vaporization of water can be considered 'vapor pressure', but wouldn't that only mean that the vapor pressure has merely risen? Why should that affect the boiling point at all, when the 'surrounding pressure' itself is still the same? Does a change in vapor pressure also affect the boiling point in any way?

After a bit of thinking, I also started wondering if the expansion of the air inside the cooker alone was enough to increase the pressure sufficiently, but I am not really too sure about it and I couldn't find much clarity on why this pressure increase occurs otherwise.

• So when the pressure cooker hits 100C all the water should turn to vapor. Now what? Given the volume difference of liquid:vapor is roughly 1:1000, unless you put very little water in the pot it simply can't fit all the vapor. Oct 5 at 13:23
• Partly related: Minimum amount of water in pressure cooker Oct 5 at 16:51

$$T_\mathrm{boil}=f(p_\mathrm{air}+p_\mathrm{water}(T))$$