Theorem relexpfldd

Description: The field of an exponentiation of a relation a subset of the relation's field. (Contributed by Drahflow, 12-Nov-2015) (Revised by RP, 30-May-2020)

	Ref	Expression
Hypothesis	relexpfldd.2	$$⊢\phi \to R\in \mathrm{V}$$
Assertion	relexpfldd	$$⊢\phi \to \left(N\in {\mathbb{N}}_0\to \bigcup \bigcup \left(R{\uparrow }_{r}N\right)\subseteq \bigcup \bigcup R\right)$$

Proof

Step	Hyp	Ref	Expression
1		relexpfldd.2	$$⊢\phi \to R\in \mathrm{V}$$
2		relexpfld	$$⊢\left(N\in {\mathbb{N}}_0\wedge R\in \mathrm{V}\right)\to \bigcup \bigcup \left(R{\uparrow }_{r}N\right)\subseteq \bigcup \bigcup R$$
3	2	ex	$$⊢N\in {\mathbb{N}}_0\to \left(R\in \mathrm{V}\to \bigcup \bigcup \left(R{\uparrow }_{r}N\right)\subseteq \bigcup \bigcup R\right)$$
4	1 3	syl5com	$$⊢\phi \to \left(N\in {\mathbb{N}}_0\to \bigcup \bigcup \left(R{\uparrow }_{r}N\right)\subseteq \bigcup \bigcup R\right)$$