Theorem brxrncnvep

Description: The range product with converse epsilon relation. (Contributed by Peter Mazsa, 22-Jun-2020) (Revised by Peter Mazsa, 22-Nov-2025)

	Ref	Expression
Assertion	brxrncnvep	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋 ) → ( 𝐴 ( 𝑅 ⋉ ◡ E ) ⟨ 𝐵 , 𝐶 ⟩ ↔ ( 𝐶 ∈ 𝐴 ∧ 𝐴 𝑅 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		brxrn	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋 ) → ( 𝐴 ( 𝑅 ⋉ ◡ E ) ⟨ 𝐵 , 𝐶 ⟩ ↔ ( 𝐴 𝑅 𝐵 ∧ 𝐴 ◡ E 𝐶 ) ) )
2		brcnvep	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐴 ◡ E 𝐶 ↔ 𝐶 ∈ 𝐴 ) )
3	2	anbi1cd	⊢ ( 𝐴 ∈ 𝑉 → ( ( 𝐴 𝑅 𝐵 ∧ 𝐴 ◡ E 𝐶 ) ↔ ( 𝐶 ∈ 𝐴 ∧ 𝐴 𝑅 𝐵 ) ) )
4	3	3ad2ant1	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋 ) → ( ( 𝐴 𝑅 𝐵 ∧ 𝐴 ◡ E 𝐶 ) ↔ ( 𝐶 ∈ 𝐴 ∧ 𝐴 𝑅 𝐵 ) ) )
5	1 4	bitrd	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋 ) → ( 𝐴 ( 𝑅 ⋉ ◡ E ) ⟨ 𝐵 , 𝐶 ⟩ ↔ ( 𝐶 ∈ 𝐴 ∧ 𝐴 𝑅 𝐵 ) ) )