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	$$⊢\phi \to R\in \mathrm{V}$$
Assertion	fvrtrcllb1d	$$⊢\phi \to R\subseteq \mathrm{t*}\left(R\right)$$

Proof

Step	Hyp	Ref	Expression
1		fvrtrcllb1d.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		nn0ex	$$⊢{\mathbb{N}}_0\in \mathrm{V}$$
4	3	a1i	$$⊢\phi \to {\mathbb{N}}_0\in \mathrm{V}$$
5		1nn0	$$⊢1\in {\mathbb{N}}_0$$
6	5	a1i	$$⊢\phi \to 1\in {\mathbb{N}}_0$$
7	2 1 4 6	fvmptiunrelexplb1d	$$⊢\phi \to R\subseteq \mathrm{t*}\left(R\right)$$