Step |
Hyp |
Ref |
Expression |
1 |
|
prdsbasmpt2.y |
⊢ 𝑌 = ( 𝑆 Xs ( 𝑥 ∈ 𝐼 ↦ 𝑅 ) ) |
2 |
|
prdsbasmpt2.b |
⊢ 𝐵 = ( Base ‘ 𝑌 ) |
3 |
|
prdsbasmpt2.s |
⊢ ( 𝜑 → 𝑆 ∈ 𝑉 ) |
4 |
|
prdsbasmpt2.i |
⊢ ( 𝜑 → 𝐼 ∈ 𝑊 ) |
5 |
|
prdsbasmpt2.r |
⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐼 𝑅 ∈ 𝑋 ) |
6 |
|
prdsbasmpt2.k |
⊢ 𝐾 = ( Base ‘ 𝑅 ) |
7 |
|
eqid |
⊢ ( 𝑥 ∈ 𝐼 ↦ 𝑅 ) = ( 𝑥 ∈ 𝐼 ↦ 𝑅 ) |
8 |
7
|
fnmpt |
⊢ ( ∀ 𝑥 ∈ 𝐼 𝑅 ∈ 𝑋 → ( 𝑥 ∈ 𝐼 ↦ 𝑅 ) Fn 𝐼 ) |
9 |
5 8
|
syl |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐼 ↦ 𝑅 ) Fn 𝐼 ) |
10 |
1 2 3 4 9
|
prdsbas2 |
⊢ ( 𝜑 → 𝐵 = X 𝑦 ∈ 𝐼 ( Base ‘ ( ( 𝑥 ∈ 𝐼 ↦ 𝑅 ) ‘ 𝑦 ) ) ) |
11 |
|
nfcv |
⊢ Ⅎ 𝑥 Base |
12 |
|
nffvmpt1 |
⊢ Ⅎ 𝑥 ( ( 𝑥 ∈ 𝐼 ↦ 𝑅 ) ‘ 𝑦 ) |
13 |
11 12
|
nffv |
⊢ Ⅎ 𝑥 ( Base ‘ ( ( 𝑥 ∈ 𝐼 ↦ 𝑅 ) ‘ 𝑦 ) ) |
14 |
|
nfcv |
⊢ Ⅎ 𝑦 ( Base ‘ ( ( 𝑥 ∈ 𝐼 ↦ 𝑅 ) ‘ 𝑥 ) ) |
15 |
|
2fveq3 |
⊢ ( 𝑦 = 𝑥 → ( Base ‘ ( ( 𝑥 ∈ 𝐼 ↦ 𝑅 ) ‘ 𝑦 ) ) = ( Base ‘ ( ( 𝑥 ∈ 𝐼 ↦ 𝑅 ) ‘ 𝑥 ) ) ) |
16 |
13 14 15
|
cbvixp |
⊢ X 𝑦 ∈ 𝐼 ( Base ‘ ( ( 𝑥 ∈ 𝐼 ↦ 𝑅 ) ‘ 𝑦 ) ) = X 𝑥 ∈ 𝐼 ( Base ‘ ( ( 𝑥 ∈ 𝐼 ↦ 𝑅 ) ‘ 𝑥 ) ) |
17 |
10 16
|
eqtrdi |
⊢ ( 𝜑 → 𝐵 = X 𝑥 ∈ 𝐼 ( Base ‘ ( ( 𝑥 ∈ 𝐼 ↦ 𝑅 ) ‘ 𝑥 ) ) ) |
18 |
7
|
fvmpt2 |
⊢ ( ( 𝑥 ∈ 𝐼 ∧ 𝑅 ∈ 𝑋 ) → ( ( 𝑥 ∈ 𝐼 ↦ 𝑅 ) ‘ 𝑥 ) = 𝑅 ) |
19 |
18
|
fveq2d |
⊢ ( ( 𝑥 ∈ 𝐼 ∧ 𝑅 ∈ 𝑋 ) → ( Base ‘ ( ( 𝑥 ∈ 𝐼 ↦ 𝑅 ) ‘ 𝑥 ) ) = ( Base ‘ 𝑅 ) ) |
20 |
19 6
|
eqtr4di |
⊢ ( ( 𝑥 ∈ 𝐼 ∧ 𝑅 ∈ 𝑋 ) → ( Base ‘ ( ( 𝑥 ∈ 𝐼 ↦ 𝑅 ) ‘ 𝑥 ) ) = 𝐾 ) |
21 |
20
|
ralimiaa |
⊢ ( ∀ 𝑥 ∈ 𝐼 𝑅 ∈ 𝑋 → ∀ 𝑥 ∈ 𝐼 ( Base ‘ ( ( 𝑥 ∈ 𝐼 ↦ 𝑅 ) ‘ 𝑥 ) ) = 𝐾 ) |
22 |
|
ixpeq2 |
⊢ ( ∀ 𝑥 ∈ 𝐼 ( Base ‘ ( ( 𝑥 ∈ 𝐼 ↦ 𝑅 ) ‘ 𝑥 ) ) = 𝐾 → X 𝑥 ∈ 𝐼 ( Base ‘ ( ( 𝑥 ∈ 𝐼 ↦ 𝑅 ) ‘ 𝑥 ) ) = X 𝑥 ∈ 𝐼 𝐾 ) |
23 |
5 21 22
|
3syl |
⊢ ( 𝜑 → X 𝑥 ∈ 𝐼 ( Base ‘ ( ( 𝑥 ∈ 𝐼 ↦ 𝑅 ) ‘ 𝑥 ) ) = X 𝑥 ∈ 𝐼 𝐾 ) |
24 |
17 23
|
eqtrd |
⊢ ( 𝜑 → 𝐵 = X 𝑥 ∈ 𝐼 𝐾 ) |