Theorem fnbrovb

Description: Value of a binary operation expressed as a binary relation. See also fnbrfvb for functions on Cartesian products. (Contributed by BJ, 15-Feb-2022)

	Ref	Expression
Assertion	fnbrovb	$$⊢\left(FFn\left(V\times W\right)\wedge \left(A\in V\wedge B\in W\right)\right)\to \left(AFB=C\leftrightarrow ⟨A,B⟩FC\right)$$

Proof

Step	Hyp	Ref	Expression
1		df-ov	$$⊢AFB=F\left(⟨A,B⟩\right)$$
2	1	eqeq1i	$$⊢AFB=C\leftrightarrow F\left(⟨A,B⟩\right)=C$$
3		fnbrfvb2	$$⊢\left(FFn\left(V\times W\right)\wedge \left(A\in V\wedge B\in W\right)\right)\to \left(F\left(⟨A,B⟩\right)=C\leftrightarrow ⟨A,B⟩FC\right)$$
4	2 3	bitrid	$$⊢\left(FFn\left(V\times W\right)\wedge \left(A\in V\wedge B\in W\right)\right)\to \left(AFB=C\leftrightarrow ⟨A,B⟩FC\right)$$