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Metamath Proof Explorer

Theorem xrninxp

Description: Intersection of a range Cartesian product with a Cartesian product. (Contributed by Peter Mazsa, 7-Apr-2020)

		Ref	Expression
	Assertion	xrninxp	⊢ ( ( 𝑅 ⋉ 𝑆 ) ∩ ( 𝐴 × ( 𝐵 × 𝐶 ) ) ) = ^◡ { ⟨ ⟨ 𝑦 , 𝑧 ⟩ , 𝑢 ⟩ ∣ ( ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶 ) ∧ ( 𝑢 ∈ 𝐴 ∧ 𝑢 ( 𝑅 ⋉ 𝑆 ) ⟨ 𝑦 , 𝑧 ⟩ ) ) }

Proof

Step	Hyp	Ref	Expression
1		inxp2	⊢ ( ( 𝑅 ⋉ 𝑆 ) ∩ ( 𝐴 × ( 𝐵 × 𝐶 ) ) ) = { ⟨ 𝑢 , 𝑥 ⟩ ∣ ( ( 𝑢 ∈ 𝐴 ∧ 𝑥 ∈ ( 𝐵 × 𝐶 ) ) ∧ 𝑢 ( 𝑅 ⋉ 𝑆 ) 𝑥 ) }
2		df-3an	⊢ ( ( 𝑢 ∈ 𝐴 ∧ 𝑥 ∈ ( 𝐵 × 𝐶 ) ∧ 𝑢 ( 𝑅 ⋉ 𝑆 ) 𝑥 ) ↔ ( ( 𝑢 ∈ 𝐴 ∧ 𝑥 ∈ ( 𝐵 × 𝐶 ) ) ∧ 𝑢 ( 𝑅 ⋉ 𝑆 ) 𝑥 ) )
3		3anan12	⊢ ( ( 𝑢 ∈ 𝐴 ∧ 𝑥 ∈ ( 𝐵 × 𝐶 ) ∧ 𝑢 ( 𝑅 ⋉ 𝑆 ) 𝑥 ) ↔ ( 𝑥 ∈ ( 𝐵 × 𝐶 ) ∧ ( 𝑢 ∈ 𝐴 ∧ 𝑢 ( 𝑅 ⋉ 𝑆 ) 𝑥 ) ) )
4	2 3	bitr3i	⊢ ( ( ( 𝑢 ∈ 𝐴 ∧ 𝑥 ∈ ( 𝐵 × 𝐶 ) ) ∧ 𝑢 ( 𝑅 ⋉ 𝑆 ) 𝑥 ) ↔ ( 𝑥 ∈ ( 𝐵 × 𝐶 ) ∧ ( 𝑢 ∈ 𝐴 ∧ 𝑢 ( 𝑅 ⋉ 𝑆 ) 𝑥 ) ) )
5	4	opabbii	⊢ { ⟨ 𝑢 , 𝑥 ⟩ ∣ ( ( 𝑢 ∈ 𝐴 ∧ 𝑥 ∈ ( 𝐵 × 𝐶 ) ) ∧ 𝑢 ( 𝑅 ⋉ 𝑆 ) 𝑥 ) } = { ⟨ 𝑢 , 𝑥 ⟩ ∣ ( 𝑥 ∈ ( 𝐵 × 𝐶 ) ∧ ( 𝑢 ∈ 𝐴 ∧ 𝑢 ( 𝑅 ⋉ 𝑆 ) 𝑥 ) ) }
6	1 5	eqtri	⊢ ( ( 𝑅 ⋉ 𝑆 ) ∩ ( 𝐴 × ( 𝐵 × 𝐶 ) ) ) = { ⟨ 𝑢 , 𝑥 ⟩ ∣ ( 𝑥 ∈ ( 𝐵 × 𝐶 ) ∧ ( 𝑢 ∈ 𝐴 ∧ 𝑢 ( 𝑅 ⋉ 𝑆 ) 𝑥 ) ) }
7		cnvopab	⊢ ^◡ { ⟨ 𝑥 , 𝑢 ⟩ ∣ ( 𝑥 ∈ ( 𝐵 × 𝐶 ) ∧ ( 𝑢 ∈ 𝐴 ∧ 𝑢 ( 𝑅 ⋉ 𝑆 ) 𝑥 ) ) } = { ⟨ 𝑢 , 𝑥 ⟩ ∣ ( 𝑥 ∈ ( 𝐵 × 𝐶 ) ∧ ( 𝑢 ∈ 𝐴 ∧ 𝑢 ( 𝑅 ⋉ 𝑆 ) 𝑥 ) ) }
8		breq2	⊢ ( 𝑥 = ⟨ 𝑦 , 𝑧 ⟩ → ( 𝑢 ( 𝑅 ⋉ 𝑆 ) 𝑥 ↔ 𝑢 ( 𝑅 ⋉ 𝑆 ) ⟨ 𝑦 , 𝑧 ⟩ ) )
9	8	anbi2d	⊢ ( 𝑥 = ⟨ 𝑦 , 𝑧 ⟩ → ( ( 𝑢 ∈ 𝐴 ∧ 𝑢 ( 𝑅 ⋉ 𝑆 ) 𝑥 ) ↔ ( 𝑢 ∈ 𝐴 ∧ 𝑢 ( 𝑅 ⋉ 𝑆 ) ⟨ 𝑦 , 𝑧 ⟩ ) ) )
10	9	dfoprab4	⊢ { ⟨ 𝑥 , 𝑢 ⟩ ∣ ( 𝑥 ∈ ( 𝐵 × 𝐶 ) ∧ ( 𝑢 ∈ 𝐴 ∧ 𝑢 ( 𝑅 ⋉ 𝑆 ) 𝑥 ) ) } = { ⟨ ⟨ 𝑦 , 𝑧 ⟩ , 𝑢 ⟩ ∣ ( ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶 ) ∧ ( 𝑢 ∈ 𝐴 ∧ 𝑢 ( 𝑅 ⋉ 𝑆 ) ⟨ 𝑦 , 𝑧 ⟩ ) ) }
11	10	cnveqi	⊢ ^◡ { ⟨ 𝑥 , 𝑢 ⟩ ∣ ( 𝑥 ∈ ( 𝐵 × 𝐶 ) ∧ ( 𝑢 ∈ 𝐴 ∧ 𝑢 ( 𝑅 ⋉ 𝑆 ) 𝑥 ) ) } = ^◡ { ⟨ ⟨ 𝑦 , 𝑧 ⟩ , 𝑢 ⟩ ∣ ( ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶 ) ∧ ( 𝑢 ∈ 𝐴 ∧ 𝑢 ( 𝑅 ⋉ 𝑆 ) ⟨ 𝑦 , 𝑧 ⟩ ) ) }
12	6 7 11	3eqtr2i	⊢ ( ( 𝑅 ⋉ 𝑆 ) ∩ ( 𝐴 × ( 𝐵 × 𝐶 ) ) ) = ^◡ { ⟨ ⟨ 𝑦 , 𝑧 ⟩ , 𝑢 ⟩ ∣ ( ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶 ) ∧ ( 𝑢 ∈ 𝐴 ∧ 𝑢 ( 𝑅 ⋉ 𝑆 ) ⟨ 𝑦 , 𝑧 ⟩ ) ) }