Lemma 14.21.5. Let $U$ be a simplicial set. Let $n \geq 0$ be an integer. The morphism $i_{n!} \text{sk}_ n U \to U$ identifies $i_{n!} \text{sk}_ n U$ with the simplicial set $U' \subset U$ defined in Lemma 14.18.4.

**Proof.**
By Lemma 14.21.4 the only nondegenerate simplices of $i_{n!} \text{sk}_ n U$ are in degrees $\leq n$. The map $i_{n!} \text{sk}_ n U \to U$ is an isomorphism in degrees $\leq n$. Combined we conclude that the map $i_{n!} \text{sk}_ n U \to U$ maps nondegenerate simplices to nondegenerate simplices and no two nondegenerate simplices have the same image. Hence Lemma 14.18.3 applies. Thus $i_{n!} \text{sk}_ n U \to U$ is injective. The result follows easily from this.
$\square$

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