Theorem aovovn0oveq

Description: If the operation's value at an argument is not the empty set, it equals the value of the alternative operation at this argument. (Contributed by Alexander van der Vekens, 26-May-2017)

	Ref	Expression
Assertion	aovovn0oveq	⊢ ( ( 𝐴 𝐹 𝐵 ) ≠ ∅ → (( 𝐴 𝐹 𝐵 )) = ( 𝐴 𝐹 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		df-ov	⊢ ( 𝐴 𝐹 𝐵 ) = ( 𝐹 ‘ ⟨ 𝐴 , 𝐵 ⟩ )
2	1	neeq1i	⊢ ( ( 𝐴 𝐹 𝐵 ) ≠ ∅ ↔ ( 𝐹 ‘ ⟨ 𝐴 , 𝐵 ⟩ ) ≠ ∅ )
3		afvfvn0fveq	⊢ ( ( 𝐹 ‘ ⟨ 𝐴 , 𝐵 ⟩ ) ≠ ∅ → ( 𝐹 ''' ⟨ 𝐴 , 𝐵 ⟩ ) = ( 𝐹 ‘ ⟨ 𝐴 , 𝐵 ⟩ ) )
4		df-aov	⊢ (( 𝐴 𝐹 𝐵 )) = ( 𝐹 ''' ⟨ 𝐴 , 𝐵 ⟩ )
5	3 4 1	3eqtr4g	⊢ ( ( 𝐹 ‘ ⟨ 𝐴 , 𝐵 ⟩ ) ≠ ∅ → (( 𝐴 𝐹 𝐵 )) = ( 𝐴 𝐹 𝐵 ) )
6	2 5	sylbi	⊢ ( ( 𝐴 𝐹 𝐵 ) ≠ ∅ → (( 𝐴 𝐹 𝐵 )) = ( 𝐴 𝐹 𝐵 ) )