Metamath Proof Explorer

Theorem fnotaovb

Description: Equivalence of operation value and ordered triple membership, analogous to fnopfvb . (Contributed by Alexander van der Vekens, 26-May-2017)

		Ref	Expression
	Assertion	fnotaovb	$⊢ (F Fn (A \times B) \land C \in A \land D \in B) \to (((C F D)) = R \leftrightarrow ⟨C, D, R⟩ \in F)$

Proof

Step	Hyp	Ref	Expression
1		opelxpi	$⊢ (C \in A \land D \in B) \to ⟨C, D⟩ \in (A \times B)$
2		fnopafvb	$⊢ (F Fn (A \times B) \land ⟨C, D⟩ \in (A \times B)) \to ((F''' ⟨C, D⟩) = R \leftrightarrow ⟨⟨C, D⟩, R⟩ \in F)$
3	1 2	sylan2	$⊢ (F Fn (A \times B) \land (C \in A \land D \in B)) \to ((F''' ⟨C, D⟩) = R \leftrightarrow ⟨⟨C, D⟩, R⟩ \in F)$
4	3	3impb	$⊢ (F Fn (A \times B) \land C \in A \land D \in B) \to ((F''' ⟨C, D⟩) = R \leftrightarrow ⟨⟨C, D⟩, R⟩ \in F)$
5		df-aov	$⊢ ((C F D)) = (F''' ⟨C, D⟩)$
6	5	eqeq1i	$⊢ ((C F D)) = R \leftrightarrow (F''' ⟨C, D⟩) = R$
7		df-ot	$⊢ ⟨C, D, R⟩ = ⟨⟨C, D⟩, R⟩$
8	7	eleq1i	$⊢ ⟨C, D, R⟩ \in F \leftrightarrow ⟨⟨C, D⟩, R⟩ \in F$
9	4 6 8	3bitr4g	$⊢ (F Fn (A \times B) \land C \in A \land D \in B) \to (((C F D)) = R \leftrightarrow ⟨C, D, R⟩ \in F)$