Theorem fvrtrcllb1d

Description: A set is a subset of its image under the reflexive-transitive closure. (Contributed by RP, 22-Jul-2020)

	Ref	Expression
Hypothesis	fvrtrcllb1d.r	⊢ ( 𝜑 → 𝑅 ∈ V )
Assertion	fvrtrcllb1d	⊢ ( 𝜑 → 𝑅 ⊆ ( t* ‘ 𝑅 ) )

Proof

Step	Hyp	Ref	Expression
1		fvrtrcllb1d.r	⊢ ( 𝜑 → 𝑅 ∈ V )
2		dfrtrcl3	⊢ t* = ( 𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 ( 𝑟 ↑𝑟 𝑛 ) )
3		nn0ex	⊢ ℕ0 ∈ V
4	3	a1i	⊢ ( 𝜑 → ℕ0 ∈ V )
5		1nn0	⊢ 1 ∈ ℕ0
6	5	a1i	⊢ ( 𝜑 → 1 ∈ ℕ0 )
7	2 1 4 6	fvmptiunrelexplb1d	⊢ ( 𝜑 → 𝑅 ⊆ ( t* ‘ 𝑅 ) )