The answer to this question at Crypto.SE is clear about what a trapdoor function is, but what makes a trapdoor function "injective"?
A trapdoor one-way injection is a trapdoor one-way function such that two different elements
of a single domain are never mapped to the same element of the corresponding codomain.
By Goldreich-Levin, if there exists an trapdoor one-way injection
then semantically secure public key encryption exists.