elrefrelsrel

Description: For sets, being an element of the class of reflexive relations ( df-refrels ) is equivalent to satisfying the reflexive relation predicate. (Contributed by Peter Mazsa, 25-Jul-2021)

		Ref	Expression
	Assertion	elrefrelsrel	$⊢ R \in V \to (R \in RefRels \leftrightarrow RefRel R)$

Proof

Step	Hyp	Ref	Expression
1		elrelsrel	$⊢ R \in V \to (R \in Rels \leftrightarrow Rel R)$
2	1	anbi2d	$⊢ R \in V \to ((I \cap (dom R \times ran R) \subseteq R \land R \in Rels) \leftrightarrow (I \cap (dom R \times ran R) \subseteq R \land Rel R))$
3		elrefrels2	$⊢ R \in RefRels \leftrightarrow (I \cap (dom R \times ran R) \subseteq R \land R \in Rels)$
4		dfrefrel2	$⊢ RefRel R \leftrightarrow (I \cap (dom R \times ran R) \subseteq R \land Rel R)$
5	2 3 4	3bitr4g	$⊢ R \in V \to (R \in RefRels \leftrightarrow RefRel R)$

Metamath Proof Explorer

Theorem elrefrelsrel

Proof