Somewhat earlier  correctly observed that "a citation for that formal definition of equality with domain but not codomain could be useful in that discussion".  The citation I gave is from Tom Apostol Calculus, page 54, but this is rather old.  Looking in recent literature was very instructive.  The problem is that definitions of function equality in the literature seem rare somehow (not found in about a dozen of recent math and CS books where such a definition could be expected).  Googling on "function equality" or "equal functions" was equally ineffective, but "functions f and g are equal" yields many results (mostly lecture notes).  Some definitions of function equality include codomain equality, some do not. Here are a few that do not include codomain equality, as requested:

http://solitaryroad.com/c303.html
 http://www.stanford.edu/class/cs103/handouts/25%20Functions.pdf
 http://www.bus.ucf.edu/sgerking/ECO%203401/Chapter%202.ppt
 http://www-math.cudenver.edu/~wcherowi/courses/m3000/lecture8a.pdf
 http://www.cse.unl.edu/~cse235/files/Functions.ppt

An interesting subtlety: many of the latter group define a codomain, but do not include it in the definition of function equality.  As Leibniz would say, this means that they do not really consider the codomain as part of the definition of a function.  More subtle: some talk about "the domain of f" but about "the codomain" (without "of f").

A pragmatic observation: authors including a codomain (say Y) in the definition of a function seem to do very little with it, except (i) defining when a function is "onto" instead of "onto Y", (ii) allowing it to restrict flexibility in the way functions can be combined, for instance, by composition.