Theorem foov

Description: An onto mapping of an operation expressed in terms of operation values. (Contributed by NM, 29-Oct-2006)

	Ref	Expression
Assertion	foov	⊢ ( 𝐹 : ( 𝐴 × 𝐵 ) –onto→ 𝐶 ↔ ( 𝐹 : ( 𝐴 × 𝐵 ) ⟶ 𝐶 ∧ ∀ 𝑧 ∈ 𝐶 ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝑧 = ( 𝑥 𝐹 𝑦 ) ) )

Proof

Step	Hyp	Ref	Expression
1		dffo3	⊢ ( 𝐹 : ( 𝐴 × 𝐵 ) –onto→ 𝐶 ↔ ( 𝐹 : ( 𝐴 × 𝐵 ) ⟶ 𝐶 ∧ ∀ 𝑧 ∈ 𝐶 ∃ 𝑤 ∈ ( 𝐴 × 𝐵 ) 𝑧 = ( 𝐹 ‘ 𝑤 ) ) )
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	eqeq2d	⊢ ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ → ( 𝑧 = ( 𝐹 ‘ 𝑤 ) ↔ 𝑧 = ( 𝑥 𝐹 𝑦 ) ) )
6	5	rexxp	⊢ ( ∃ 𝑤 ∈ ( 𝐴 × 𝐵 ) 𝑧 = ( 𝐹 ‘ 𝑤 ) ↔ ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝑧 = ( 𝑥 𝐹 𝑦 ) )
7	6	ralbii	⊢ ( ∀ 𝑧 ∈ 𝐶 ∃ 𝑤 ∈ ( 𝐴 × 𝐵 ) 𝑧 = ( 𝐹 ‘ 𝑤 ) ↔ ∀ 𝑧 ∈ 𝐶 ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝑧 = ( 𝑥 𝐹 𝑦 ) )
8	7	anbi2i	⊢ ( ( 𝐹 : ( 𝐴 × 𝐵 ) ⟶ 𝐶 ∧ ∀ 𝑧 ∈ 𝐶 ∃ 𝑤 ∈ ( 𝐴 × 𝐵 ) 𝑧 = ( 𝐹 ‘ 𝑤 ) ) ↔ ( 𝐹 : ( 𝐴 × 𝐵 ) ⟶ 𝐶 ∧ ∀ 𝑧 ∈ 𝐶 ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝑧 = ( 𝑥 𝐹 𝑦 ) ) )
9	1 8	bitri	⊢ ( 𝐹 : ( 𝐴 × 𝐵 ) –onto→ 𝐶 ↔ ( 𝐹 : ( 𝐴 × 𝐵 ) ⟶ 𝐶 ∧ ∀ 𝑧 ∈ 𝐶 ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝑧 = ( 𝑥 𝐹 𝑦 ) ) )