Theorem fovrn

Description: An operation's value belongs to its codomain. (Contributed by NM, 27-Aug-2006)

	Ref	Expression
Assertion	fovrn	⊢ ( ( 𝐹 : ( 𝑅 × 𝑆 ) ⟶ 𝐶 ∧ 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) → ( 𝐴 𝐹 𝐵 ) ∈ 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		opelxpi	⊢ ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) → ⟨ 𝐴 , 𝐵 ⟩ ∈ ( 𝑅 × 𝑆 ) )
2		df-ov	⊢ ( 𝐴 𝐹 𝐵 ) = ( 𝐹 ‘ ⟨ 𝐴 , 𝐵 ⟩ )
3		ffvelrn	⊢ ( ( 𝐹 : ( 𝑅 × 𝑆 ) ⟶ 𝐶 ∧ ⟨ 𝐴 , 𝐵 ⟩ ∈ ( 𝑅 × 𝑆 ) ) → ( 𝐹 ‘ ⟨ 𝐴 , 𝐵 ⟩ ) ∈ 𝐶 )
4	2 3	eqeltrid	⊢ ( ( 𝐹 : ( 𝑅 × 𝑆 ) ⟶ 𝐶 ∧ ⟨ 𝐴 , 𝐵 ⟩ ∈ ( 𝑅 × 𝑆 ) ) → ( 𝐴 𝐹 𝐵 ) ∈ 𝐶 )
5	1 4	sylan2	⊢ ( ( 𝐹 : ( 𝑅 × 𝑆 ) ⟶ 𝐶 ∧ ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) ) → ( 𝐴 𝐹 𝐵 ) ∈ 𝐶 )
6	5	3impb	⊢ ( ( 𝐹 : ( 𝑅 × 𝑆 ) ⟶ 𝐶 ∧ 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) → ( 𝐴 𝐹 𝐵 ) ∈ 𝐶 )