This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

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 ) )