We can go further, actually. We could argue even that (y, f) is not, actually, an image and that's it's just a number and a function. Rather, the image is what's on the screen and only exists when it's displayed.
I'm not trying to continue an argument, but rather just point out that the philosophy of identity can be an elusive thing.
To make things even more fun:
Take an image x, decimate it, and obtain a lower-resolution version of it, which we call z. Are x and z the same image?
Mathematically, no. But when displayed on the screen, then most people would say they are the same. Of course, it depends on the definition of is, but such discussion would be utterly pointless. If we use the same definition, then there's no ambiguity.
Such lack of precision is only tolerated because computation is fast and cheap. I would love to see people trying all f mappings with the help of an abacus! Let's be thankful that we live in this golden age.