Theorem relexprel

Description: The exponentiation of a relation is a relation. (Contributed by RP, 23-May-2020)

	Ref	Expression
Assertion	relexprel	$$⊢\left(N\in {\mathbb{N}}_0\wedge R\in V\wedge \mathrm{Rel}R\right)\to \mathrm{Rel}\left(R{\uparrow }_{r}N\right)$$

Proof

Step	Hyp	Ref	Expression
1		ax-1	$$⊢\mathrm{Rel}R\to \left(N=1\to \mathrm{Rel}R\right)$$
2		relexprelg	$$⊢\left(N\in {\mathbb{N}}_0\wedge R\in V\wedge \left(N=1\to \mathrm{Rel}R\right)\right)\to \mathrm{Rel}\left(R{\uparrow }_{r}N\right)$$
3	1 2	syl3an3	$$⊢\left(N\in {\mathbb{N}}_0\wedge R\in V\wedge \mathrm{Rel}R\right)\to \mathrm{Rel}\left(R{\uparrow }_{r}N\right)$$