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 |
|
prdsbasmpt.t |
⊢ ( 𝜑 → 𝑇 ∈ 𝐵 ) |
7 |
|
prdsbasprj.j |
⊢ ( 𝜑 → 𝐽 ∈ 𝐼 ) |
8 |
|
fveq2 |
⊢ ( 𝑥 = 𝐽 → ( 𝑇 ‘ 𝑥 ) = ( 𝑇 ‘ 𝐽 ) ) |
9 |
|
2fveq3 |
⊢ ( 𝑥 = 𝐽 → ( Base ‘ ( 𝑅 ‘ 𝑥 ) ) = ( Base ‘ ( 𝑅 ‘ 𝐽 ) ) ) |
10 |
8 9
|
eleq12d |
⊢ ( 𝑥 = 𝐽 → ( ( 𝑇 ‘ 𝑥 ) ∈ ( Base ‘ ( 𝑅 ‘ 𝑥 ) ) ↔ ( 𝑇 ‘ 𝐽 ) ∈ ( Base ‘ ( 𝑅 ‘ 𝐽 ) ) ) ) |
11 |
1 2 3 4 5
|
prdsbas2 |
⊢ ( 𝜑 → 𝐵 = X 𝑥 ∈ 𝐼 ( Base ‘ ( 𝑅 ‘ 𝑥 ) ) ) |
12 |
6 11
|
eleqtrd |
⊢ ( 𝜑 → 𝑇 ∈ X 𝑥 ∈ 𝐼 ( Base ‘ ( 𝑅 ‘ 𝑥 ) ) ) |
13 |
|
elixp2 |
⊢ ( 𝑇 ∈ X 𝑥 ∈ 𝐼 ( Base ‘ ( 𝑅 ‘ 𝑥 ) ) ↔ ( 𝑇 ∈ V ∧ 𝑇 Fn 𝐼 ∧ ∀ 𝑥 ∈ 𝐼 ( 𝑇 ‘ 𝑥 ) ∈ ( Base ‘ ( 𝑅 ‘ 𝑥 ) ) ) ) |
14 |
13
|
simp3bi |
⊢ ( 𝑇 ∈ X 𝑥 ∈ 𝐼 ( Base ‘ ( 𝑅 ‘ 𝑥 ) ) → ∀ 𝑥 ∈ 𝐼 ( 𝑇 ‘ 𝑥 ) ∈ ( Base ‘ ( 𝑅 ‘ 𝑥 ) ) ) |
15 |
12 14
|
syl |
⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐼 ( 𝑇 ‘ 𝑥 ) ∈ ( Base ‘ ( 𝑅 ‘ 𝑥 ) ) ) |
16 |
10 15 7
|
rspcdva |
⊢ ( 𝜑 → ( 𝑇 ‘ 𝐽 ) ∈ ( Base ‘ ( 𝑅 ‘ 𝐽 ) ) ) |