Suppose A and B are sets such that card A ≤ card B. Prove that there exists a set C ⊆ B such that card C = card A.
I know that there is only an injection from A to B. I'm having trouble showing that set C which is a subset of B has the same cardinality as set A.
Since $\text{card}(A)\leq \text{card}(B)$, there is an injection $f\colon A\to B$. Then $f\colon A\to f(A)$ is a bijection, and $f(A)\subseteq B$.