Theorem cosscnvssid5

Description: Equivalent expressions for the class of cosets by the converse of the relation R to be a subset of the identity class. (Contributed by Peter Mazsa, 5-Sep-2021)

	Ref	Expression
Assertion	cosscnvssid5	⊢ ( ( ≀ ◡ 𝑅 ⊆ I ∧ Rel 𝑅 ) ↔ ( ∀ 𝑢 ∈ dom 𝑅 ∀ 𝑣 ∈ dom 𝑅 ( 𝑢 = 𝑣 ∨ ( [ 𝑢 ] 𝑅 ∩ [ 𝑣 ] 𝑅 ) = ∅ ) ∧ Rel 𝑅 ) )

Proof

Step	Hyp	Ref	Expression
1		cosscnvssid4	⊢ ( ≀ ◡ 𝑅 ⊆ I ↔ ∀ 𝑥 ∃* 𝑢 𝑢 𝑅 𝑥 )
2	1	anbi1i	⊢ ( ( ≀ ◡ 𝑅 ⊆ I ∧ Rel 𝑅 ) ↔ ( ∀ 𝑥 ∃* 𝑢 𝑢 𝑅 𝑥 ∧ Rel 𝑅 ) )
3		inecmo3	⊢ ( ( ∀ 𝑢 ∈ dom 𝑅 ∀ 𝑣 ∈ dom 𝑅 ( 𝑢 = 𝑣 ∨ ( [ 𝑢 ] 𝑅 ∩ [ 𝑣 ] 𝑅 ) = ∅ ) ∧ Rel 𝑅 ) ↔ ( ∀ 𝑥 ∃* 𝑢 𝑢 𝑅 𝑥 ∧ Rel 𝑅 ) )
4	2 3	bitr4i	⊢ ( ( ≀ ◡ 𝑅 ⊆ I ∧ Rel 𝑅 ) ↔ ( ∀ 𝑢 ∈ dom 𝑅 ∀ 𝑣 ∈ dom 𝑅 ( 𝑢 = 𝑣 ∨ ( [ 𝑢 ] 𝑅 ∩ [ 𝑣 ] 𝑅 ) = ∅ ) ∧ Rel 𝑅 ) )