Theorem relmpoopab

Description: Any function to sets of ordered pairs produces a relation on function value unconditionally. (Contributed by Mario Carneiro, 9-Feb-2015)

	Ref	Expression
Hypothesis	relmpoopab.1	⊢ 𝐹 = ( 𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵 ↦ { ⟨ 𝑧 , 𝑤 ⟩ ∣ 𝜑 } )
Assertion	relmpoopab	⊢ Rel ( 𝐶 𝐹 𝐷 )

Proof

Step	Hyp	Ref	Expression
1		relmpoopab.1	⊢ 𝐹 = ( 𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵 ↦ { ⟨ 𝑧 , 𝑤 ⟩ ∣ 𝜑 } )
2		relopabv	⊢ Rel { ⟨ 𝑧 , 𝑤 ⟩ ∣ 𝜑 }
3		df-rel	⊢ ( Rel { ⟨ 𝑧 , 𝑤 ⟩ ∣ 𝜑 } ↔ { ⟨ 𝑧 , 𝑤 ⟩ ∣ 𝜑 } ⊆ ( V × V ) )
4	2 3	mpbi	⊢ { ⟨ 𝑧 , 𝑤 ⟩ ∣ 𝜑 } ⊆ ( V × V )
5	4	rgen2w	⊢ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 { ⟨ 𝑧 , 𝑤 ⟩ ∣ 𝜑 } ⊆ ( V × V )
6	1	ovmptss	⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 { ⟨ 𝑧 , 𝑤 ⟩ ∣ 𝜑 } ⊆ ( V × V ) → ( 𝐶 𝐹 𝐷 ) ⊆ ( V × V ) )
7	5 6	ax-mp	⊢ ( 𝐶 𝐹 𝐷 ) ⊆ ( V × V )
8		df-rel	⊢ ( Rel ( 𝐶 𝐹 𝐷 ) ↔ ( 𝐶 𝐹 𝐷 ) ⊆ ( V × V ) )
9	7 8	mpbir	⊢ Rel ( 𝐶 𝐹 𝐷 )