Can I be getting an YLOD from a faulty ppower supply?

So, my PS3 got the YLOD some days ago, but it didn't actually "break down". What happens is that every time that I try to play it, the YLOD shows up, the fan doesn't spin and then the red light flashes. Then, I press the power button again and the red light stops flashing. So, I press the power button one more time and the PS3 works normally: I can play games and watch movies as any other PS3 would. I've heard around the internet (I don''t remember the website URL) that if you have an YLOD and you don't hear the fan spinning up before the actual yellow light flashes, you may have a faulty power supply (PSU). Is this correct? The main reason for me to consider it being a power supply problem is that back in the old days when I played an unpatched Assassin's Creed ( due to the fact that I didn't have an internet connection ) the game would crash a lot as this was a common issue: see this link. Therefore, when the game crashed, I had to turn the system down using the switch in the back. This is what is causing me to think that the PSU is the problem, I'm considering that I kinda broke it by turning the system down so many times with the switch after the game crashed.

Thanks in advance,

Pedro.