Metamath Proof Explorer

Theorem 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 ∈ CnvRefRels ↔ ≀ R ⊆ I ∧ ≀ R ∈ Rels

Proof

Step Hyp Ref Expression
1 elcnvrefrels2 ⊢ ≀ R ∈ CnvRefRels ↔ ≀ R ⊆ I ∩ dom ⁡ ≀ R × ran ⁡ ≀ R ∧ ≀ R ∈ Rels
2 cossssid ⊢ ≀ R ⊆ I ↔ ≀ R ⊆ I ∩ dom ⁡ ≀ R × ran ⁡ ≀ R
3 2 anbi1i ⊢ ≀ R ⊆ I ∧ ≀ R ∈ Rels ↔ ≀ R ⊆ I ∩ dom ⁡ ≀ R × ran ⁡ ≀ R ∧ ≀ R ∈ Rels
4 1 3 bitr4i ⊢ ≀ R ∈ CnvRefRels ↔ ≀ R ⊆ I ∧ ≀ R ∈ Rels