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 ) )