Theorem aovvoveq

Description: The alternative value of the operation on an ordered pair equals the operation's value on this ordered pair. (Contributed by Alexander van der Vekens, 26-May-2017)

	Ref	Expression
Assertion	aovvoveq	⊢ ( (( 𝐴 𝐹 𝐵 )) ∈ 𝐶 → (( 𝐴 𝐹 𝐵 )) = ( 𝐴 𝐹 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		df-aov	⊢ (( 𝐴 𝐹 𝐵 )) = ( 𝐹 ''' ⟨ 𝐴 , 𝐵 ⟩ )
2	1	eleq1i	⊢ ( (( 𝐴 𝐹 𝐵 )) ∈ 𝐶 ↔ ( 𝐹 ''' ⟨ 𝐴 , 𝐵 ⟩ ) ∈ 𝐶 )
3		afvvfveq	⊢ ( ( 𝐹 ''' ⟨ 𝐴 , 𝐵 ⟩ ) ∈ 𝐶 → ( 𝐹 ''' ⟨ 𝐴 , 𝐵 ⟩ ) = ( 𝐹 ‘ ⟨ 𝐴 , 𝐵 ⟩ ) )
4		df-ov	⊢ ( 𝐴 𝐹 𝐵 ) = ( 𝐹 ‘ ⟨ 𝐴 , 𝐵 ⟩ )
5	3 1 4	3eqtr4g	⊢ ( ( 𝐹 ''' ⟨ 𝐴 , 𝐵 ⟩ ) ∈ 𝐶 → (( 𝐴 𝐹 𝐵 )) = ( 𝐴 𝐹 𝐵 ) )
6	2 5	sylbi	⊢ ( (( 𝐴 𝐹 𝐵 )) ∈ 𝐶 → (( 𝐴 𝐹 𝐵 )) = ( 𝐴 𝐹 𝐵 ) )