Theorem elixpconst

Description: Membership in an infinite Cartesian product of a constant B . (Contributed by NM, 12-Apr-2008)

	Ref	Expression
Hypothesis	elixp.1	$$⊢F\in \mathrm{V}$$
Assertion	elixpconst	$$⊢F\in \underset{x\in A}{⨉}B\leftrightarrow F:A⟶B$$

Proof

Step	Hyp	Ref	Expression
1		elixp.1	$$⊢F\in \mathrm{V}$$
2	1	elixp	$$⊢F\in \underset{x\in A}{⨉}B\leftrightarrow \left(FFnA\wedge \forall x\in AF\left(x\right)\in B\right)$$
3		ffnfv	$$⊢F:A⟶B\leftrightarrow \left(FFnA\wedge \forall x\in AF\left(x\right)\in B\right)$$
4	2 3	bitr4i	$$⊢F\in \underset{x\in A}{⨉}B\leftrightarrow F:A⟶B$$