Theorem ixpssmap

Description: An infinite Cartesian product is a subset of set exponentiation. Remark in Enderton p. 54. (Contributed by NM, 28-Sep-2006)

	Ref	Expression
Hypothesis	ixpssmap.2	⊢ 𝐵 ∈ V
Assertion	ixpssmap	⊢ X 𝑥 ∈ 𝐴 𝐵 ⊆ ( ∪ 𝑥 ∈ 𝐴 𝐵 ↑m 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		ixpssmap.2	⊢ 𝐵 ∈ V
2	1	rgenw	⊢ ∀ 𝑥 ∈ 𝐴 𝐵 ∈ V
3		ixpssmapg	⊢ ( ∀ 𝑥 ∈ 𝐴 𝐵 ∈ V → X 𝑥 ∈ 𝐴 𝐵 ⊆ ( ∪ 𝑥 ∈ 𝐴 𝐵 ↑m 𝐴 ) )
4	2 3	ax-mp	⊢ X 𝑥 ∈ 𝐴 𝐵 ⊆ ( ∪ 𝑥 ∈ 𝐴 𝐵 ↑m 𝐴 )