Proving the existence of a bijection.

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Hello all,

I've recently used a property that seems perfectly valid, yet upon further scrutiny I could not come up with a way to prove it. Here is what I would like some help on.

Given two sets X and Y and functions f and g mapping X into Y, with the property that f is injective and g is surjective, prove there exists a bijection from X into Y.

I believe this has to be true for the following reason. Since bijections create an equivalence relation wrt cardinality, we can think of injections as saying Y is no smaller than X, and surjections as saying that Y is no bigger than X. Together we have that Y is is the same size as X, and hence they are equal cardinality. Therefore there exists a bijection between the two.

However, this heuristic argument aside, I can not think of a proof of this proposition.
 

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