cardinality of finite subset

Question

I have the following question: Prove that a set A has the same cardinality of a subset of a Set B, if and only if exists an injective function A to B.

I find it hard to prove it because I can easily find a set A which is:

A={1,2}
B={1,2,3,4}
C={1,2,3}

C is a subset of B. C's cardinality is bigger than A's.

There is an injective function from A to B, of course. Assunption: A,B,C are finite. where am I wrong?

Thank you.