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) (Revised by AV, 12-Jul-2024)

	Ref	Expression
Hypothesis	relexpfldd.1	$$⊢\phi \to N\in {\mathbb{N}}_0$$
Assertion	relexpfldd	$$⊢\phi \to \bigcup \bigcup \left(R{\uparrow }_{r}N\right)\subseteq \bigcup \bigcup R$$

Proof

Step	Hyp	Ref	Expression
1		relexpfldd.1	$$⊢\phi \to N\in {\mathbb{N}}_0$$
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	1 2	sylan	$$⊢\left(\phi \wedge R\in \mathrm{V}\right)\to \bigcup \bigcup \left(R{\uparrow }_{r}N\right)\subseteq \bigcup \bigcup R$$
4	3	ex	$$⊢\phi \to \left(R\in \mathrm{V}\to \bigcup \bigcup \left(R{\uparrow }_{r}N\right)\subseteq \bigcup \bigcup R\right)$$
5		reldmrelexp	$$⊢\mathrm{Rel}\mathrm{dom}{\uparrow }_r$$
6	5	ovprc1	$$⊢\neg R\in \mathrm{V}\to R{\uparrow }_{r}N=\varnothing$$
7	6	unieqd	$$⊢\neg R\in \mathrm{V}\to \bigcup \left(R{\uparrow }_{r}N\right)=\bigcup \varnothing$$
8		uni0	$$⊢\bigcup \varnothing =\varnothing$$
9	7 8	eqtrdi	$$⊢\neg R\in \mathrm{V}\to \bigcup \left(R{\uparrow }_{r}N\right)=\varnothing$$
10	9	unieqd	$$⊢\neg R\in \mathrm{V}\to \bigcup \bigcup \left(R{\uparrow }_{r}N\right)=\bigcup \varnothing$$
11	10 8	eqtrdi	$$⊢\neg R\in \mathrm{V}\to \bigcup \bigcup \left(R{\uparrow }_{r}N\right)=\varnothing$$
12		0ss	$$⊢\varnothing \subseteq \bigcup \bigcup R$$
13	11 12	eqsstrdi	$$⊢\neg R\in \mathrm{V}\to \bigcup \bigcup \left(R{\uparrow }_{r}N\right)\subseteq \bigcup \bigcup R$$
14	4 13	pm2.61d1	$$⊢\phi \to \bigcup \bigcup \left(R{\uparrow }_{r}N\right)\subseteq \bigcup \bigcup R$$