Metamath Proof Explorer

Theorem cosselcnvrefrels3

Description: Necessary and sufficient condition for a coset relation to be an element of the converse reflexive relation class. (Contributed by Peter Mazsa, 30-Aug-2021)

		Ref	Expression
	Assertion	cosselcnvrefrels3	$⊢ ≀ R \in CnvRefRels \leftrightarrow (\forall u \forall x \forall y ((u R x \land u R y) \to x = y) \land ≀ R \in Rels)$

Proof

Step	Hyp	Ref	Expression
1		cosselcnvrefrels2	$⊢ ≀ R \in CnvRefRels \leftrightarrow (≀ R \subseteq I \land ≀ R \in Rels)$
2		cossssid3	$⊢ ≀ R \subseteq I \leftrightarrow \forall u \forall x \forall y ((u R x \land u R y) \to x = y)$
3	2	anbi1i	$⊢ (≀ R \subseteq I \land ≀ R \in Rels) \leftrightarrow (\forall u \forall x \forall y ((u R x \land u R y) \to x = y) \land ≀ R \in Rels)$
4	1 3	bitri	$⊢ ≀ R \in CnvRefRels \leftrightarrow (\forall u \forall x \forall y ((u R x \land u R y) \to x = y) \land ≀ R \in Rels)$