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	$$⊢((AFB))\in C\to ((AFB))=AFB$$

Proof

Step	Hyp	Ref	Expression
1		df-aov	$$⊢((AFB))=\left(F\mathrm{\text{'}\text{'}\text{'}}⟨A,B⟩\right)$$
2	1	eleq1i	$$⊢((AFB))\in C\leftrightarrow \left(F\mathrm{\text{'}\text{'}\text{'}}⟨A,B⟩\right)\in C$$
3		afvvfveq	$$⊢\left(F\mathrm{\text{'}\text{'}\text{'}}⟨A,B⟩\right)\in C\to \left(F\mathrm{\text{'}\text{'}\text{'}}⟨A,B⟩\right)=F\left(⟨A,B⟩\right)$$
4		df-ov	$$⊢AFB=F\left(⟨A,B⟩\right)$$
5	3 1 4	3eqtr4g	$$⊢\left(F\mathrm{\text{'}\text{'}\text{'}}⟨A,B⟩\right)\in C\to ((AFB))=AFB$$
6	2 5	sylbi	$$⊢((AFB))\in C\to ((AFB))=AFB$$