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	$$⊢\phi \to R\in \mathrm{V}$$
Assertion	fvrtrcllb0d	$$⊢\phi \to {\mathrm{I}\upharpoonright }_{\left(\mathrm{dom}R\cup \mathrm{ran}R\right)}\subseteq \mathrm{t*}\left(R\right)$$

Proof

Step	Hyp	Ref	Expression
1		fvrtrcllb0d.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		0nn0	$$⊢0\in {\mathbb{N}}_0$$
6	5	a1i	$$⊢\phi \to 0\in {\mathbb{N}}_0$$
7	2 1 4 6	fvmptiunrelexplb0d	$$⊢\phi \to {\mathrm{I}\upharpoonright }_{\left(\mathrm{dom}R\cup \mathrm{ran}R\right)}\subseteq \mathrm{t*}\left(R\right)$$