Theorem elrtrclrec

Description: Membership in the indexed union of relation exponentiation over the natural numbers (including zero) is equivalent to the existence of at least one number such that the element is a member of that relationship power. (Contributed by RP, 2-Jun-2020)

	Ref	Expression
Hypothesis	rtrclrec.def	$$⊢C=\left(r\in \mathrm{V}⟼\bigcup _{n\in {\mathbb{N}}_0}\left(r{\uparrow }_{r}n\right)\right)$$
Assertion	elrtrclrec	$$⊢R\in V\to \left(X\in C\left(R\right)\leftrightarrow \exists n\in {\mathbb{N}}_{0}X\in \left(R{\uparrow }_{r}n\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		rtrclrec.def	$$⊢C=\left(r\in \mathrm{V}⟼\bigcup _{n\in {\mathbb{N}}_0}\left(r{\uparrow }_{r}n\right)\right)$$
2		nn0ex	$$⊢{\mathbb{N}}_0\in \mathrm{V}$$
3	1	eliunov2	$$⊢\left(R\in V\wedge {\mathbb{N}}_0\in \mathrm{V}\right)\to \left(X\in C\left(R\right)\leftrightarrow \exists n\in {\mathbb{N}}_{0}X\in \left(R{\uparrow }_{r}n\right)\right)$$
4	2 3	mpan2	$$⊢R\in V\to \left(X\in C\left(R\right)\leftrightarrow \exists n\in {\mathbb{N}}_{0}X\in \left(R{\uparrow }_{r}n\right)\right)$$