Theorem elec1cnvxrn2

Description: Elementhood in the converse range Cartesian product coset of A . (Contributed by Peter Mazsa, 11-Jul-2021)

	Ref	Expression
Assertion	elec1cnvxrn2	⊢ ( 𝐵 ∈ 𝑉 → ( 𝐵 ∈ [ 𝐴 ] ◡ ( 𝑅 ⋉ 𝑆 ) ↔ ∃ 𝑦 ∃ 𝑧 ( 𝐴 = ⟨ 𝑦 , 𝑧 ⟩ ∧ 𝐵 𝑅 𝑦 ∧ 𝐵 𝑆 𝑧 ) ) )

Proof

Step	Hyp	Ref	Expression
1		relcnv	⊢ 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		br1cnvxrn2	⊢ ( 𝐵 ∈ 𝑉 → ( 𝐴 ◡ ( 𝑅 ⋉ 𝑆 ) 𝐵 ↔ ∃ 𝑦 ∃ 𝑧 ( 𝐴 = ⟨ 𝑦 , 𝑧 ⟩ ∧ 𝐵 𝑅 𝑦 ∧ 𝐵 𝑆 𝑧 ) ) )
5	3 4	bitrid	⊢ ( 𝐵 ∈ 𝑉 → ( 𝐵 ∈ [ 𝐴 ] ◡ ( 𝑅 ⋉ 𝑆 ) ↔ ∃ 𝑦 ∃ 𝑧 ( 𝐴 = ⟨ 𝑦 , 𝑧 ⟩ ∧ 𝐵 𝑅 𝑦 ∧ 𝐵 𝑆 𝑧 ) ) )