Theorem fvrtrcllb0d

Description: A restriction of the identity relation is a subset of the reflexive-transitive closure of a set. (Contributed by RP, 22-Jul-2020)

	Ref	Expression
Hypothesis	fvrtrcllb0d.r	⊢ ( 𝜑 → 𝑅 ∈ V )
Assertion	fvrtrcllb0d	⊢ ( 𝜑 → ( I ↾ ( dom 𝑅 ∪ ran 𝑅 ) ) ⊆ ( t* ‘ 𝑅 ) )

Proof

Step	Hyp	Ref	Expression
1		fvrtrcllb0d.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		0nn0	⊢ 0 ∈ ℕ0
6	5	a1i	⊢ ( 𝜑 → 0 ∈ ℕ0 )
7	2 1 4 6	fvmptiunrelexplb0d	⊢ ( 𝜑 → ( I ↾ ( dom 𝑅 ∪ ran 𝑅 ) ) ⊆ ( t* ‘ 𝑅 ) )