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 ) )