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	$$⊢AFB\ne \varnothing \to ((AFB))=AFB$$

Proof

Step	Hyp	Ref	Expression
1		df-ov	$$⊢AFB=F\left(⟨A,B⟩\right)$$
2	1	neeq1i	$$⊢AFB\ne \varnothing \leftrightarrow F\left(⟨A,B⟩\right)\ne \varnothing$$
3		afvfvn0fveq	$$⊢F\left(⟨A,B⟩\right)\ne \varnothing \to \left(F\mathrm{\text{'}\text{'}\text{'}}⟨A,B⟩\right)=F\left(⟨A,B⟩\right)$$
4		df-aov	$$⊢((AFB))=\left(F\mathrm{\text{'}\text{'}\text{'}}⟨A,B⟩\right)$$
5	3 4 1	3eqtr4g	$$⊢F\left(⟨A,B⟩\right)\ne \varnothing \to ((AFB))=AFB$$
6	2 5	sylbi	$$⊢AFB\ne \varnothing \to ((AFB))=AFB$$