Image of a morphism in an abelian category

Hi,

In an abelian category a morphism f:A->B can be factored as A->Imf->B where the first morphism is epi and the second monic. In Weibel's book on homological algebra it's given as an "exercise for the reader" to prove that the first map is epi and it's given directly after giving the axioms hinting that there should be an easy direct proof (or that it's long and he doesn't bother to prove it...).

Now I've seen before a proof of this in MacLane's book Categories for the Working Mathematician, but this proof depends on a lot of previous theory developed in the book. My question is, does anyone know a simple short proof following directly from the axioms?

Regards,
Mats
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