Metamath Proof Explorer

Theorem cosselcnvrefrels4

Description: Necessary and sufficient condition for a coset relation to be an element of the converse reflexive relation class. (Contributed by Peter Mazsa, 31-Aug-2021)

Ref Expression
Assertion cosselcnvrefrels4 ⊢ ≀ R ∈ CnvRefRels ↔ ∀ u ∃* x u R x ∧ ≀ R ∈ Rels

Proof

Step Hyp Ref Expression
1 cosselcnvrefrels2 ⊢ ≀ R ∈ CnvRefRels ↔ ≀ R ⊆ I ∧ ≀ R ∈ Rels
2 cossssid4 ⊢ ≀ R ⊆ I ↔ ∀ u ∃* x u R x
3 2 anbi1i ⊢ ≀ R ⊆ I ∧ ≀ R ∈ Rels ↔ ∀ u ∃* x u R x ∧ ≀ R ∈ Rels
4 1 3 bitri ⊢ ≀ R ∈ CnvRefRels ↔ ∀ u ∃* x u R x ∧ ≀ R ∈ Rels