Theorem fnbrfvb2

Description: Version of fnbrfvb for functions on Cartesian products: function value expressed as a binary relation. See fnbrovb for the form when F is seen as a binary operation. (Contributed by BJ, 15-Feb-2022)

	Ref	Expression
Assertion	fnbrfvb2	$$⊢\left(FFn\left(V\times W\right)\wedge \left(A\in V\wedge B\in W\right)\right)\to \left(F\left(⟨A,B⟩\right)=C\leftrightarrow ⟨A,B⟩FC\right)$$

Proof

Step	Hyp	Ref	Expression
1		opelxpi	$$⊢\left(A\in V\wedge B\in W\right)\to ⟨A,B⟩\in \left(V\times W\right)$$
2		fnbrfvb	$$⊢\left(FFn\left(V\times W\right)\wedge ⟨A,B⟩\in \left(V\times W\right)\right)\to \left(F\left(⟨A,B⟩\right)=C\leftrightarrow ⟨A,B⟩FC\right)$$
3	1 2	sylan2	$$⊢\left(FFn\left(V\times W\right)\wedge \left(A\in V\wedge B\in W\right)\right)\to \left(F\left(⟨A,B⟩\right)=C\leftrightarrow ⟨A,B⟩FC\right)$$