There's something wrong here. First of all there is nothing in your statement to force A to be at least countably infinite, which is a requirement for C to be countably infinite.

For a counter-example let A = {1, 2, 3} and B = {1, 2, 3, 4, 5, 6, 7}. Define an injection . Then and f is an injection, but there exists no set that is countably infinite.

-Dan