Theorem ixpfn

Description: A nuple is a function. (Contributed by FL, 6-Jun-2011) (Revised by Mario Carneiro, 31-May-2014)

	Ref	Expression
Assertion	ixpfn	$$⊢F\in \underset{x\in A}{⨉}B\to FFnA$$

Proof

Step	Hyp	Ref	Expression
1		fneq1	$$⊢f=F\to \left(fFnA\leftrightarrow FFnA\right)$$
2		elixp2	$$⊢f\in \underset{x\in A}{⨉}B\leftrightarrow \left(f\in \mathrm{V}\wedge fFnA\wedge \forall x\in Af\left(x\right)\in B\right)$$
3	2	simp2bi	$$⊢f\in \underset{x\in A}{⨉}B\to fFnA$$
4	1 3	vtoclga	$$⊢F\in \underset{x\in A}{⨉}B\to FFnA$$