Theorem xrninxp2

Description: Intersection of a range Cartesian product with a Cartesian product. (Contributed by Peter Mazsa, 8-Apr-2020)

	Ref	Expression
Assertion	xrninxp2	⊢ ( ( 𝑅 ⋉ 𝑆 ) ∩ ( 𝐴 × ( 𝐵 × 𝐶 ) ) ) = { ⟨ 𝑢 , 𝑥 ⟩ ∣ ( 𝑥 ∈ ( 𝐵 × 𝐶 ) ∧ ( 𝑢 ∈ 𝐴 ∧ 𝑢 ( 𝑅 ⋉ 𝑆 ) 𝑥 ) ) }

Proof

Step	Hyp	Ref	Expression
1		inxp2	⊢ ( ( 𝑅 ⋉ 𝑆 ) ∩ ( 𝐴 × ( 𝐵 × 𝐶 ) ) ) = { ⟨ 𝑢 , 𝑥 ⟩ ∣ ( ( 𝑢 ∈ 𝐴 ∧ 𝑥 ∈ ( 𝐵 × 𝐶 ) ) ∧ 𝑢 ( 𝑅 ⋉ 𝑆 ) 𝑥 ) }
2		an21	⊢ ( ( ( 𝑢 ∈ 𝐴 ∧ 𝑥 ∈ ( 𝐵 × 𝐶 ) ) ∧ 𝑢 ( 𝑅 ⋉ 𝑆 ) 𝑥 ) ↔ ( 𝑥 ∈ ( 𝐵 × 𝐶 ) ∧ ( 𝑢 ∈ 𝐴 ∧ 𝑢 ( 𝑅 ⋉ 𝑆 ) 𝑥 ) ) )
3	2	opabbii	⊢ { ⟨ 𝑢 , 𝑥 ⟩ ∣ ( ( 𝑢 ∈ 𝐴 ∧ 𝑥 ∈ ( 𝐵 × 𝐶 ) ) ∧ 𝑢 ( 𝑅 ⋉ 𝑆 ) 𝑥 ) } = { ⟨ 𝑢 , 𝑥 ⟩ ∣ ( 𝑥 ∈ ( 𝐵 × 𝐶 ) ∧ ( 𝑢 ∈ 𝐴 ∧ 𝑢 ( 𝑅 ⋉ 𝑆 ) 𝑥 ) ) }
4	1 3	eqtri	⊢ ( ( 𝑅 ⋉ 𝑆 ) ∩ ( 𝐴 × ( 𝐵 × 𝐶 ) ) ) = { ⟨ 𝑢 , 𝑥 ⟩ ∣ ( 𝑥 ∈ ( 𝐵 × 𝐶 ) ∧ ( 𝑢 ∈ 𝐴 ∧ 𝑢 ( 𝑅 ⋉ 𝑆 ) 𝑥 ) ) }