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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 ∈ CnvRefRels ↔ ∀ u ∀ x ∀ y u R x ∧ u R y → x = y ∧ ≀ R ∈ Rels

Proof

Step Hyp Ref Expression
1 cosselcnvrefrels2 ⊢ ≀ R ∈ CnvRefRels ↔ ≀ R ⊆ I ∧ ≀ R ∈ Rels
2 cossssid3 ⊢ ≀ R ⊆ I ↔ ∀ u ∀ x ∀ y u R x ∧ u R y → x = y
3 2 anbi1i ⊢ ≀ R ⊆ I ∧ ≀ R ∈ Rels ↔ ∀ u ∀ x ∀ y u R x ∧ u R y → x = y ∧ ≀ R ∈ Rels
4 1 3 bitri ⊢ ≀ R ∈ CnvRefRels ↔ ∀ u ∀ x ∀ y u R x ∧ u R y → x = y ∧ ≀ R ∈ Rels