Theorem ffnov

Description: An operation maps to a class to which all values belong. (Contributed by NM, 7-Feb-2004)

	Ref	Expression
Assertion	ffnov	⊢ ( 𝐹 : ( 𝐴 × 𝐵 ) ⟶ 𝐶 ↔ ( 𝐹 Fn ( 𝐴 × 𝐵 ) ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ( 𝑥 𝐹 𝑦 ) ∈ 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		ffnfv	⊢ ( 𝐹 : ( 𝐴 × 𝐵 ) ⟶ 𝐶 ↔ ( 𝐹 Fn ( 𝐴 × 𝐵 ) ∧ ∀ 𝑤 ∈ ( 𝐴 × 𝐵 ) ( 𝐹 ‘ 𝑤 ) ∈ 𝐶 ) )
2		fveq2	⊢ ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ → ( 𝐹 ‘ 𝑤 ) = ( 𝐹 ‘ ⟨ 𝑥 , 𝑦 ⟩ ) )
3		df-ov	⊢ ( 𝑥 𝐹 𝑦 ) = ( 𝐹 ‘ ⟨ 𝑥 , 𝑦 ⟩ )
4	2 3	eqtr4di	⊢ ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ → ( 𝐹 ‘ 𝑤 ) = ( 𝑥 𝐹 𝑦 ) )
5	4	eleq1d	⊢ ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ → ( ( 𝐹 ‘ 𝑤 ) ∈ 𝐶 ↔ ( 𝑥 𝐹 𝑦 ) ∈ 𝐶 ) )
6	5	ralxp	⊢ ( ∀ 𝑤 ∈ ( 𝐴 × 𝐵 ) ( 𝐹 ‘ 𝑤 ) ∈ 𝐶 ↔ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ( 𝑥 𝐹 𝑦 ) ∈ 𝐶 )
7	6	anbi2i	⊢ ( ( 𝐹 Fn ( 𝐴 × 𝐵 ) ∧ ∀ 𝑤 ∈ ( 𝐴 × 𝐵 ) ( 𝐹 ‘ 𝑤 ) ∈ 𝐶 ) ↔ ( 𝐹 Fn ( 𝐴 × 𝐵 ) ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ( 𝑥 𝐹 𝑦 ) ∈ 𝐶 ) )
8	1 7	bitri	⊢ ( 𝐹 : ( 𝐴 × 𝐵 ) ⟶ 𝐶 ↔ ( 𝐹 Fn ( 𝐴 × 𝐵 ) ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ( 𝑥 𝐹 𝑦 ) ∈ 𝐶 ) )