I recently had problem set which asked to prove that if for any , we have

then is an injection. (EDITED**)

My proof started with the intention to show that if (pointwise) . And I did that, but when I got it returned, apparently I didn't use the definition of an injection since injectivity implies that for all implies .

However, I feel that my proof was valid since A and B can have very well been singleton sets with x1 and x2, respectively. Is my approach incorrect?