Mind does not appear to exist during deep sleep, in coma, in certain brain deceases*/injuries, under general anesthesia etc. So, one can say that if brain does not work (or works partially), mind does not exist (or does not behave normally) and hence mind is entirely due to brain. However, if we assume that entire mind is due to brain it is not possible to explain a) near-death experiences that some people have b) out of body experiences that some people have and c) research findings on reincarnation by Dr. Ian Stevenson. These experiences/findings point towards the phenomenon of mind (or faculty of mind – whatever you like to call it like soul etc.) getting detached from the body.
If we assume for a moment that the mind is separate from the body let us see how we can explain why mind does not appear to exist during deep sleep, in coma, in certain brain diseases*/injuries, under general anesthesia etc. Mind interprets information sent by the brain. If brain is inactive/damaged/dead, this information is not sent properly (or not sent at all) to the mind for processing which makes it look as if mind is not working properly or is not functioning at all. For analogy sake consider a TV (mind) connected to a cable (brain/body) and to a power supply. If there is no signal sent by the cable, the TV does not work. This does not mean that the TV is not in working condition. This explains how the dependence of mind on brain is possible even if mind does not exist due to brain.
Spiritually speaking, consciousness is big container in which there is mind inside of which there is body and brain is part of the body. Mind is the interface between the consciousness and the body. Mind receives lot of information from the body/brain and tunes in to specific information and projects it on the screen of consciousness.
*In certain brain diseases the person does not behave normally and one says that mind is not normal.