1. **State the problem:** We want to show that in a substitution cipher, where no two letters are sent to the same letter, every letter appears exactly once as an image of a letter. 2. **Understand the definition:** A substitution cipher is a function from the set of letters (say, the alphabet) to itself such that no two distinct letters map to the same letter. This means the function is injective (one-to-one). 3. **Show that every letter appears once:** Since the alphabet has a finite number of letters (say $n$), and the function is injective from a set of size $n$ to itself, it must also be surjective (onto). This is because an injective function from a finite set to itself cannot miss any element; otherwise, the function would not be injective. 4. **Formal reasoning:** Let the alphabet be $A$ with $|A|=n$. The substitution cipher is a function $f:A \to A$ that is injective. By the pigeonhole principle, an injective function from a finite set to itself is also surjective. 5. **Conclusion:** Therefore, every letter in the alphabet appears exactly once as an image of some letter under the substitution cipher. It is impossible to miss any letter, such as the letter $y$, because that would mean the function is not surjective and hence not injective. **Final answer:** In a substitution cipher, every letter appears once and only once as an image of a letter, so missing the letter $y$ is impossible.