Metamath Proof Explorer

Theorem ixpf

Description: A member of an infinite Cartesian product maps to the indexed union of the product argument. Remark in Enderton p. 54. (Contributed by NM, 28-Sep-2006)

		Ref	Expression
	Assertion	ixpf	$⊢ F \in \underset{x \in A}{⨉} B \to F : A ⟶ ⋃_{x \in A} B$

Proof

Step	Hyp	Ref	Expression
1		elixp2	$⊢ F \in \underset{x \in A}{⨉} B \leftrightarrow (F \in V \land F Fn A \land \forall x \in A F (x) \in B)$
2		ssiun2	$⊢ x \in A \to B \subseteq ⋃_{x \in A} B$
3	2	sseld	$⊢ x \in A \to (F (x) \in B \to F (x) \in ⋃_{x \in A} B)$
4	3	ralimia	$⊢ \forall x \in A F (x) \in B \to \forall x \in A F (x) \in ⋃_{x \in A} B$
5	4	anim2i	$⊢ (F Fn A \land \forall x \in A F (x) \in B) \to (F Fn A \land \forall x \in A F (x) \in ⋃_{x \in A} B)$
6		nfcv	$⊢ \underline{Ⅎ} x A$
7		nfiu1	$⊢ \underline{Ⅎ} x ⋃_{x \in A} B$
8		nfcv	$⊢ \underline{Ⅎ} x F$
9	6 7 8	ffnfvf	$⊢ F : A ⟶ ⋃_{x \in A} B \leftrightarrow (F Fn A \land \forall x \in A F (x) \in ⋃_{x \in A} B)$
10	5 9	sylibr	$⊢ (F Fn A \land \forall x \in A F (x) \in B) \to F : A ⟶ ⋃_{x \in A} B$
11	10	3adant1	$⊢ (F \in V \land F Fn A \land \forall x \in A F (x) \in B) \to F : A ⟶ ⋃_{x \in A} B$
12	1 11	sylbi	$⊢ F \in \underset{x \in A}{⨉} B \to F : A ⟶ ⋃_{x \in A} B$