Metamath Proof Explorer
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 ) |