Theorem ovres

Description: The value of a restricted operation. (Contributed by FL, 10-Nov-2006)

	Ref	Expression
Assertion	ovres	⊢ ( ( 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ) → ( 𝐴 ( 𝐹 ↾ ( 𝐶 × 𝐷 ) ) 𝐵 ) = ( 𝐴 𝐹 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		opelxpi	⊢ ( ( 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ) → ⟨ 𝐴 , 𝐵 ⟩ ∈ ( 𝐶 × 𝐷 ) )
2	1	fvresd	⊢ ( ( 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ) → ( ( 𝐹 ↾ ( 𝐶 × 𝐷 ) ) ‘ ⟨ 𝐴 , 𝐵 ⟩ ) = ( 𝐹 ‘ ⟨ 𝐴 , 𝐵 ⟩ ) )
3		df-ov	⊢ ( 𝐴 ( 𝐹 ↾ ( 𝐶 × 𝐷 ) ) 𝐵 ) = ( ( 𝐹 ↾ ( 𝐶 × 𝐷 ) ) ‘ ⟨ 𝐴 , 𝐵 ⟩ )
4		df-ov	⊢ ( 𝐴 𝐹 𝐵 ) = ( 𝐹 ‘ ⟨ 𝐴 , 𝐵 ⟩ )
5	2 3 4	3eqtr4g	⊢ ( ( 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ) → ( 𝐴 ( 𝐹 ↾ ( 𝐶 × 𝐷 ) ) 𝐵 ) = ( 𝐴 𝐹 𝐵 ) )