If there is an onto function $f: A\rightarrow B$ how to prove $\#(B)\leq\#(A)$?

Question

If there is a bijection function between two sets, they have the same cardinality. But if this function is onto? I know that a bijection function is an onto function, so for an equal signal, it is immediately defined. But how to prove it could be less too?

Answer 1

Hint: An onto function has a section (thanks to the axiom of choice),i.e. there exists a function $s:B\longrightarrow A$ such that$$f\circ s=\operatorname{id}_B,$$ so that $s$ is one-to-one.