Theorem cosscnv

Description: Class of cosets by the converse of R (Contributed by Peter Mazsa, 17-Jun-2020)

	Ref	Expression
Assertion	cosscnv	⊢ ≀ ◡ 𝑅 = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑢 ( 𝑥 𝑅 𝑢 ∧ 𝑦 𝑅 𝑢 ) }

Proof

Step	Hyp	Ref	Expression
1		df-coss	⊢ ≀ ◡ 𝑅 = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑢 ( 𝑢 ◡ 𝑅 𝑥 ∧ 𝑢 ◡ 𝑅 𝑦 ) }
2		brcnvg	⊢ ( ( 𝑢 ∈ V ∧ 𝑥 ∈ V ) → ( 𝑢 ◡ 𝑅 𝑥 ↔ 𝑥 𝑅 𝑢 ) )
3	2	el2v	⊢ ( 𝑢 ◡ 𝑅 𝑥 ↔ 𝑥 𝑅 𝑢 )
4		brcnvg	⊢ ( ( 𝑢 ∈ V ∧ 𝑦 ∈ V ) → ( 𝑢 ◡ 𝑅 𝑦 ↔ 𝑦 𝑅 𝑢 ) )
5	4	el2v	⊢ ( 𝑢 ◡ 𝑅 𝑦 ↔ 𝑦 𝑅 𝑢 )
6	3 5	anbi12i	⊢ ( ( 𝑢 ◡ 𝑅 𝑥 ∧ 𝑢 ◡ 𝑅 𝑦 ) ↔ ( 𝑥 𝑅 𝑢 ∧ 𝑦 𝑅 𝑢 ) )
7	6	exbii	⊢ ( ∃ 𝑢 ( 𝑢 ◡ 𝑅 𝑥 ∧ 𝑢 ◡ 𝑅 𝑦 ) ↔ ∃ 𝑢 ( 𝑥 𝑅 𝑢 ∧ 𝑦 𝑅 𝑢 ) )
8	7	opabbii	⊢ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑢 ( 𝑢 ◡ 𝑅 𝑥 ∧ 𝑢 ◡ 𝑅 𝑦 ) } = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑢 ( 𝑥 𝑅 𝑢 ∧ 𝑦 𝑅 𝑢 ) }
9	1 8	eqtri	⊢ ≀ ◡ 𝑅 = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑢 ( 𝑥 𝑅 𝑢 ∧ 𝑦 𝑅 𝑢 ) }