But in theory, if your "funnel type thing" had a suitable lens at the far (big) end that was designed to focus the light from the 10" aperture in such a way that it hits the 6" mirror (or objective lens) as parallel rays, then yes, it could increase the aperture and light-gathering power of your telescope - but what you have really done is made a bigger 10" compound telescope!
I suspect the optics of such a telescope with a large aperture objective lens and an intermediate aperture mirror / objective are not practical - or somebody would surely have already built such a beast! You can have a go at designing one using an optics simulator like this:
http://physics.bu.edu/~duffy/java/Opticsa1.html