Theorem fnov

Description: Representation of a function in terms of its values. (Contributed by NM, 7-Feb-2004) (Revised by Mario Carneiro, 31-Aug-2015)

	Ref	Expression
Assertion	fnov	$$⊢FFn\left(A\times B\right)\leftrightarrow F=\left(x\in A,y\in B⟼xFy\right)$$

Proof

Step	Hyp	Ref	Expression
1		dffn5	$$⊢FFn\left(A\times B\right)\leftrightarrow F=\left(z\in \left(A\times B\right)⟼F\left(z\right)\right)$$
2		fveq2	$$⊢z=⟨x,y⟩\to F\left(z\right)=F\left(⟨x,y⟩\right)$$
3		df-ov	$$⊢xFy=F\left(⟨x,y⟩\right)$$
4	2 3	eqtr4di	$$⊢z=⟨x,y⟩\to F\left(z\right)=xFy$$
5	4	mpompt	$$⊢\left(z\in \left(A\times B\right)⟼F\left(z\right)\right)=\left(x\in A,y\in B⟼xFy\right)$$
6	5	eqeq2i	$$⊢F=\left(z\in \left(A\times B\right)⟼F\left(z\right)\right)\leftrightarrow F=\left(x\in A,y\in B⟼xFy\right)$$
7	1 6	bitri	$$⊢FFn\left(A\times B\right)\leftrightarrow F=\left(x\in A,y\in B⟼xFy\right)$$