cosselcnvrefrels2

Description: Necessary and sufficient condition for a coset relation to be an element of the converse reflexive relation class. (Contributed by Peter Mazsa, 25-Aug-2021)

		Ref	Expression
	Assertion	cosselcnvrefrels2	$⊢ ≀ R \in CnvRefRels \leftrightarrow (≀ R \subseteq I \land ≀ R \in Rels)$

Proof

Step	Hyp	Ref	Expression
1		elcnvrefrels2	$⊢ ≀ R \in CnvRefRels \leftrightarrow (≀ R \subseteq I \cap (dom ≀ R \times ran ≀ R) \land ≀ R \in Rels)$
2		cossssid	$⊢ ≀ R \subseteq I \leftrightarrow ≀ R \subseteq I \cap (dom ≀ R \times ran ≀ R)$
3	2	anbi1i	$⊢ (≀ R \subseteq I \land ≀ R \in Rels) \leftrightarrow (≀ R \subseteq I \cap (dom ≀ R \times ran ≀ R) \land ≀ R \in Rels)$
4	1 3	bitr4i	$⊢ ≀ R \in CnvRefRels \leftrightarrow (≀ R \subseteq I \land ≀ R \in Rels)$

Metamath Proof Explorer

Theorem cosselcnvrefrels2

Proof