Theorem refrelressn

Description: Any class ' R ' restricted to the singleton of the set ' A ' (see ressn2 ) is reflexive. (Contributed by Peter Mazsa, 12-Jun-2024)

	Ref	Expression
Assertion	refrelressn	$$⊢A\in V\to RefRel\left({R\upharpoonright }_{\left\{A\right\}}\right)$$

Proof

Step	Hyp	Ref	Expression
1		refressn	$$⊢A\in V\to \forall x\in \left(\mathrm{dom}\left({R\upharpoonright }_{\left\{A\right\}}\right)\cap \mathrm{ran}\left({R\upharpoonright }_{\left\{A\right\}}\right)\right)x\left({R\upharpoonright }_{\left\{A\right\}}\right)x$$
2		relres	$$⊢\mathrm{Rel}\left({R\upharpoonright }_{\left\{A\right\}}\right)$$
3		dfrefrel5	$$⊢RefRel\left({R\upharpoonright }_{\left\{A\right\}}\right)\leftrightarrow \left(\forall x\in \left(\mathrm{dom}\left({R\upharpoonright }_{\left\{A\right\}}\right)\cap \mathrm{ran}\left({R\upharpoonright }_{\left\{A\right\}}\right)\right)x\left({R\upharpoonright }_{\left\{A\right\}}\right)x\wedge \mathrm{Rel}\left({R\upharpoonright }_{\left\{A\right\}}\right)\right)$$
4	1 2 3	sylanblrc	$$⊢A\in V\to RefRel\left({R\upharpoonright }_{\left\{A\right\}}\right)$$