Theorem cosselcnvrefrels5

Description: Necessary and sufficient condition for a coset relation to be an element of the converse reflexive relation class. (Contributed by Peter Mazsa, 5-Sep-2021)

	Ref	Expression
Assertion	cosselcnvrefrels5	⊢ ( ≀ 𝑅 ∈ CnvRefRels ↔ ( ∀ 𝑥 ∈ ran 𝑅 ∀ 𝑦 ∈ ran 𝑅 ( 𝑥 = 𝑦 ∨ ( [ 𝑥 ] ◡ 𝑅 ∩ [ 𝑦 ] ◡ 𝑅 ) = ∅ ) ∧ ≀ 𝑅 ∈ Rels ) )

Proof

Step	Hyp	Ref	Expression
1		cosselcnvrefrels2	⊢ ( ≀ 𝑅 ∈ CnvRefRels ↔ ( ≀ 𝑅 ⊆ I ∧ ≀ 𝑅 ∈ Rels ) )
2		cossssid5	⊢ ( ≀ 𝑅 ⊆ I ↔ ∀ 𝑥 ∈ ran 𝑅 ∀ 𝑦 ∈ ran 𝑅 ( 𝑥 = 𝑦 ∨ ( [ 𝑥 ] ◡ 𝑅 ∩ [ 𝑦 ] ◡ 𝑅 ) = ∅ ) )
3	2	anbi1i	⊢ ( ( ≀ 𝑅 ⊆ I ∧ ≀ 𝑅 ∈ Rels ) ↔ ( ∀ 𝑥 ∈ ran 𝑅 ∀ 𝑦 ∈ ran 𝑅 ( 𝑥 = 𝑦 ∨ ( [ 𝑥 ] ◡ 𝑅 ∩ [ 𝑦 ] ◡ 𝑅 ) = ∅ ) ∧ ≀ 𝑅 ∈ Rels ) )
4	1 3	bitri	⊢ ( ≀ 𝑅 ∈ CnvRefRels ↔ ( ∀ 𝑥 ∈ ran 𝑅 ∀ 𝑦 ∈ ran 𝑅 ( 𝑥 = 𝑦 ∨ ( [ 𝑥 ] ◡ 𝑅 ∩ [ 𝑦 ] ◡ 𝑅 ) = ∅ ) ∧ ≀ 𝑅 ∈ Rels ) )