Theorem elecxrn

Description: Elementhood in the ( R |X. S ) -coset of A . (Contributed by Peter Mazsa, 18-Apr-2020) (Revised by Peter Mazsa, 21-Sep-2021)

	Ref	Expression
Assertion	elecxrn	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐵 ∈ [ 𝐴 ] ( 𝑅 ⋉ 𝑆 ) ↔ ∃ 𝑥 ∃ 𝑦 ( 𝐵 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝐴 𝑅 𝑥 ∧ 𝐴 𝑆 𝑦 ) ) )

Proof

Step	Hyp	Ref	Expression
1		xrnrel	⊢ Rel ( 𝑅 ⋉ 𝑆 )
2		relelec	⊢ ( Rel ( 𝑅 ⋉ 𝑆 ) → ( 𝐵 ∈ [ 𝐴 ] ( 𝑅 ⋉ 𝑆 ) ↔ 𝐴 ( 𝑅 ⋉ 𝑆 ) 𝐵 ) )
3	1 2	ax-mp	⊢ ( 𝐵 ∈ [ 𝐴 ] ( 𝑅 ⋉ 𝑆 ) ↔ 𝐴 ( 𝑅 ⋉ 𝑆 ) 𝐵 )
4		brxrn2	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐴 ( 𝑅 ⋉ 𝑆 ) 𝐵 ↔ ∃ 𝑥 ∃ 𝑦 ( 𝐵 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝐴 𝑅 𝑥 ∧ 𝐴 𝑆 𝑦 ) ) )
5	3 4	syl5bb	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐵 ∈ [ 𝐴 ] ( 𝑅 ⋉ 𝑆 ) ↔ ∃ 𝑥 ∃ 𝑦 ( 𝐵 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝐴 𝑅 𝑥 ∧ 𝐴 𝑆 𝑦 ) ) )