I see two common use cases, and I will not use the word “proof” to avoid the semantic politics:
User wants to see what the image will directly look like if it gets saved using some output color space and rendering intent,
e.g. sRGB/perceptual (squashing all the colors into sRGB).
e.g. JapanColor2003WebCoated.icc/relative colorimetric.
The output color space could be a printer color space.
The color pipeline would be: RT working space > RT ouput space (could be printer) > Monitor profile
User wants to see what the image will look like if it gets saved using some output color space and rendering intent and then subsequently will be printed.
e.g. saved to sRGB then sent to a printer which uses JapanColor2003WebCoated.icc/relative colorimetric.
The output color space would not be a printer color space.
The color pipeline would be: RT working space > RT output space (not printer) > RT printer space > Monitor profile
Basically step 2 is a shortcut so one does not have to save the raw image first to TIFF using a typical output space like sRGB and then having to open the TIFF and set the printer space as output and view what that would look like.