Step |
Hyp |
Ref |
Expression |
1 |
|
prdsbasmpt.y |
⊢ 𝑌 = ( 𝑆 Xs 𝑅 ) |
2 |
|
prdsbasmpt.b |
⊢ 𝐵 = ( Base ‘ 𝑌 ) |
3 |
|
prdsbasmpt.s |
⊢ ( 𝜑 → 𝑆 ∈ 𝑉 ) |
4 |
|
prdsbasmpt.i |
⊢ ( 𝜑 → 𝐼 ∈ 𝑊 ) |
5 |
|
prdsbasmpt.r |
⊢ ( 𝜑 → 𝑅 Fn 𝐼 ) |
6 |
1 2 3 4 5
|
prdsbas2 |
⊢ ( 𝜑 → 𝐵 = X 𝑥 ∈ 𝐼 ( Base ‘ ( 𝑅 ‘ 𝑥 ) ) ) |
7 |
6
|
eleq2d |
⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐼 ↦ 𝑈 ) ∈ 𝐵 ↔ ( 𝑥 ∈ 𝐼 ↦ 𝑈 ) ∈ X 𝑥 ∈ 𝐼 ( Base ‘ ( 𝑅 ‘ 𝑥 ) ) ) ) |
8 |
|
mptelixpg |
⊢ ( 𝐼 ∈ 𝑊 → ( ( 𝑥 ∈ 𝐼 ↦ 𝑈 ) ∈ X 𝑥 ∈ 𝐼 ( Base ‘ ( 𝑅 ‘ 𝑥 ) ) ↔ ∀ 𝑥 ∈ 𝐼 𝑈 ∈ ( Base ‘ ( 𝑅 ‘ 𝑥 ) ) ) ) |
9 |
4 8
|
syl |
⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐼 ↦ 𝑈 ) ∈ X 𝑥 ∈ 𝐼 ( Base ‘ ( 𝑅 ‘ 𝑥 ) ) ↔ ∀ 𝑥 ∈ 𝐼 𝑈 ∈ ( Base ‘ ( 𝑅 ‘ 𝑥 ) ) ) ) |
10 |
7 9
|
bitrd |
⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐼 ↦ 𝑈 ) ∈ 𝐵 ↔ ∀ 𝑥 ∈ 𝐼 𝑈 ∈ ( Base ‘ ( 𝑅 ‘ 𝑥 ) ) ) ) |