Theorem br2coss

Description: Cosets by ,R binary relation. (Contributed by Peter Mazsa, 25-Aug-2019)

	Ref	Expression
Assertion	br2coss	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( 𝐴 ≀ ≀ 𝑅 𝐵 ↔ ( [ 𝐴 ] ≀ 𝑅 ∩ [ 𝐵 ] ≀ 𝑅 ) ≠ ∅ ) )

Proof

Step	Hyp	Ref	Expression
1		brcoss3	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( 𝐴 ≀ ≀ 𝑅 𝐵 ↔ ( [ 𝐴 ] ◡ ≀ 𝑅 ∩ [ 𝐵 ] ◡ ≀ 𝑅 ) ≠ ∅ ) )
2		cnvcosseq	⊢ ◡ ≀ 𝑅 = ≀ 𝑅
3	2	eceq2i	⊢ [ 𝐴 ] ◡ ≀ 𝑅 = [ 𝐴 ] ≀ 𝑅
4	2	eceq2i	⊢ [ 𝐵 ] ◡ ≀ 𝑅 = [ 𝐵 ] ≀ 𝑅
5	3 4	ineq12i	⊢ ( [ 𝐴 ] ◡ ≀ 𝑅 ∩ [ 𝐵 ] ◡ ≀ 𝑅 ) = ( [ 𝐴 ] ≀ 𝑅 ∩ [ 𝐵 ] ≀ 𝑅 )
6	5	neeq1i	⊢ ( ( [ 𝐴 ] ◡ ≀ 𝑅 ∩ [ 𝐵 ] ◡ ≀ 𝑅 ) ≠ ∅ ↔ ( [ 𝐴 ] ≀ 𝑅 ∩ [ 𝐵 ] ≀ 𝑅 ) ≠ ∅ )
7	1 6	bitrdi	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( 𝐴 ≀ ≀ 𝑅 𝐵 ↔ ( [ 𝐴 ] ≀ 𝑅 ∩ [ 𝐵 ] ≀ 𝑅 ) ≠ ∅ ) )