Theorem br1cnvxrn2

Description: The converse of a binary relation over a range Cartesian product. (Contributed by Peter Mazsa, 11-Jul-2021)

	Ref	Expression
Assertion	br1cnvxrn2	⊢ ( 𝐵 ∈ 𝑉 → ( 𝐴 ◡ ( 𝑅 ⋉ 𝑆 ) 𝐵 ↔ ∃ 𝑦 ∃ 𝑧 ( 𝐴 = ⟨ 𝑦 , 𝑧 ⟩ ∧ 𝐵 𝑅 𝑦 ∧ 𝐵 𝑆 𝑧 ) ) )

Proof

Step	Hyp	Ref	Expression
1		xrnrel	⊢ Rel ( 𝑅 ⋉ 𝑆 )
2	1	relbrcnv	⊢ ( 𝐴 ◡ ( 𝑅 ⋉ 𝑆 ) 𝐵 ↔ 𝐵 ( 𝑅 ⋉ 𝑆 ) 𝐴 )
3		brxrn2	⊢ ( 𝐵 ∈ 𝑉 → ( 𝐵 ( 𝑅 ⋉ 𝑆 ) 𝐴 ↔ ∃ 𝑦 ∃ 𝑧 ( 𝐴 = ⟨ 𝑦 , 𝑧 ⟩ ∧ 𝐵 𝑅 𝑦 ∧ 𝐵 𝑆 𝑧 ) ) )
4	2 3	syl5bb	⊢ ( 𝐵 ∈ 𝑉 → ( 𝐴 ◡ ( 𝑅 ⋉ 𝑆 ) 𝐵 ↔ ∃ 𝑦 ∃ 𝑧 ( 𝐴 = ⟨ 𝑦 , 𝑧 ⟩ ∧ 𝐵 𝑅 𝑦 ∧ 𝐵 𝑆 𝑧 ) ) )