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	⊢ ( 𝜑 → 𝑅 ∈ V )
Assertion	brfvrtrcld	⊢ ( 𝜑 → ( 𝐴 ( t* ‘ 𝑅 ) 𝐵 ↔ ∃ 𝑛 ∈ ℕ0 𝐴 ( 𝑅 ↑𝑟 𝑛 ) 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		brfvrtrcld.r	⊢ ( 𝜑 → 𝑅 ∈ V )
2		dfrtrcl3	⊢ t* = ( 𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 ( 𝑟 ↑𝑟 𝑛 ) )
3		ssidd	⊢ ( 𝜑 → ℕ0 ⊆ ℕ0 )
4	2 1 3	brmptiunrelexpd	⊢ ( 𝜑 → ( 𝐴 ( t* ‘ 𝑅 ) 𝐵 ↔ ∃ 𝑛 ∈ ℕ0 𝐴 ( 𝑅 ↑𝑟 𝑛 ) 𝐵 ) )