Im not a hundred percent sure but I will give it a try. Just because you change the F-stop doesn't mean you change the focal length. By changing focal length the fov changes but when changing the F-stop the entire image is still going through the lens and hitting the sensor. The F-stop only changes the amount of light and Dof. I know my explanation sucks and may be wrong but that is the best I could come up with.
All of the above saying is true but I didn't understand what it has (even remotely) anything to do with what I am asking

Plus I already said "Assume fixed focal length please...) so why chaging focal length. Sorry you tried but not even close...
KmH, what you are saying is true as well but it is not the answer to my question.
My question is not a cropped sensor body question. It is a generic questions that can apply to any SLR camera.
Say you have a 50mm prime lens (CAN'T change the Focal Length!..)
at f/1.0 your aperture diameter is 50mm (a huge opening). You take a picture. Then you stop down to smallest opening say f/22 (or maybe smaller doesn't matter). At f/22, all you have is an opening that is less then 3mm. All other parameters same. You take another picture and you have the SAME FOV... (don't get into DOF, I am NOT even talking about it) you get no vignetting on the sensor. Same picture on both images (with different DOF of course) but same FOV!...
This I cannot explain. The only one explanation I can come up with is that if the image circle is large enough to cover the entire image sensor at smallest aperture (the highest f number ), then it is alright. If this is true then I know that at any f-stop, I will have an image circle that will cover the entire sensor.
But if not, then I cannot explain how we get the same FOV when chaging the opening that lets the light enter and reach the image sensor(or film).
I am hoping that I am able explain myself. If not, then I will have to sketch it up next time.
So it sounds complex but actually is not when you think about it. And knowing the answer to this will not make me a better photographer. But I will feel better.
As I think about it now, I feel that I should have posted this in "Beyond the basic" forum... Any moderator reading this
