Theorem relexpdm

Description: The domain of an exponentiation of a relation a subset of the relation's field. (Contributed by RP, 23-May-2020)

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

Proof

Step	Hyp	Ref	Expression
1		relexpdmg	$$⊢\left(N\in {\mathbb{N}}_0\wedge R\in V\right)\to \mathrm{dom}\left(R{\uparrow }_{r}N\right)\subseteq \mathrm{dom}R\cup \mathrm{ran}R$$
2		dmrnssfld	$$⊢\mathrm{dom}R\cup \mathrm{ran}R\subseteq \bigcup \bigcup R$$
3	1 2	sstrdi	$$⊢\left(N\in {\mathbb{N}}_0\wedge R\in V\right)\to \mathrm{dom}\left(R{\uparrow }_{r}N\right)\subseteq \bigcup \bigcup R$$