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 ) )