Theorem fvrcllb1d

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

	Ref	Expression
Hypothesis	fvrcllb1d.r	$$⊢\phi \to R\in \mathrm{V}$$
Assertion	fvrcllb1d	$$⊢\phi \to R\subseteq \mathrm{r*}\left(R\right)$$

Proof

Step	Hyp	Ref	Expression
1		fvrcllb1d.r	$$⊢\phi \to R\in \mathrm{V}$$
2		dfrcl4	$$⊢\mathrm{r*}=\left(r\in \mathrm{V}⟼\bigcup _{n\in \left\{0,1\right\}}\left(r{\uparrow }_{r}n\right)\right)$$
3		prex	$$⊢\left\{0,1\right\}\in \mathrm{V}$$
4	3	a1i	$$⊢\phi \to \left\{0,1\right\}\in \mathrm{V}$$
5		1ex	$$⊢1\in \mathrm{V}$$
6	5	prid2	$$⊢1\in \left\{0,1\right\}$$
7	6	a1i	$$⊢\phi \to 1\in \left\{0,1\right\}$$
8	2 1 4 7	fvmptiunrelexplb1d	$$⊢\phi \to R\subseteq \mathrm{r*}\left(R\right)$$