Theorem brfvrtrcld

Description: If two elements are connected by the reflexive-transitive closure of a relation, then they are connected via n instances the relation, for some natural number n . Similar of dfrtrclrec2 . (Contributed by RP, 22-Jul-2020)

	Ref	Expression
Hypothesis	brfvrtrcld.r	$$⊢\phi \to R\in \mathrm{V}$$
Assertion	brfvrtrcld	$$⊢\phi \to \left(A\mathrm{t*}\left(R\right)B\leftrightarrow \exists n\in {\mathbb{N}}_{0}A\left(R{\uparrow }_{r}n\right)B\right)$$

Proof

Step	Hyp	Ref	Expression
1		brfvrtrcld.r	$$⊢\phi \to R\in \mathrm{V}$$
2		dfrtrcl3	$$⊢\mathrm{t*}=\left(r\in \mathrm{V}⟼\bigcup _{n\in {\mathbb{N}}_0}\left(r{\uparrow }_{r}n\right)\right)$$
3		ssidd	$$⊢\phi \to {\mathbb{N}}_0\subseteq {\mathbb{N}}_0$$
4	2 1 3	brmptiunrelexpd	$$⊢\phi \to \left(A\mathrm{t*}\left(R\right)B\leftrightarrow \exists n\in {\mathbb{N}}_{0}A\left(R{\uparrow }_{r}n\right)B\right)$$