Metamath Proof Explorer

Theorem cosselcnvrefrels5

Description: Necessary and sufficient condition for a coset relation to be an element of the converse reflexive relation class. (Contributed by Peter Mazsa, 5-Sep-2021)

Ref Expression
Assertion cosselcnvrefrels5 ⊢ ≀ R ∈ CnvRefRels ↔ ∀ x ∈ ran ⁡ R ∀ y ∈ ran ⁡ R x = y ∨ x R -1 ∩ y R -1 = ∅ ∧ ≀ R ∈ Rels

Proof

Step Hyp Ref Expression
1 cosselcnvrefrels2 ⊢ ≀ R ∈ CnvRefRels ↔ ≀ R ⊆ I ∧ ≀ R ∈ Rels
2 cossssid5 ⊢ ≀ R ⊆ I ↔ ∀ x ∈ ran ⁡ R ∀ y ∈ ran ⁡ R x = y ∨ x R -1 ∩ y R -1 = ∅
3 2 anbi1i ⊢ ≀ R ⊆ I ∧ ≀ R ∈ Rels ↔ ∀ x ∈ ran ⁡ R ∀ y ∈ ran ⁡ R x = y ∨ x R -1 ∩ y R -1 = ∅ ∧ ≀ R ∈ Rels
4 1 3 bitri ⊢ ≀ R ∈ CnvRefRels ↔ ∀ x ∈ ran ⁡ R ∀ y ∈ ran ⁡ R x = y ∨ x R -1 ∩ y R -1 = ∅ ∧ ≀ R ∈ Rels