Step |
Hyp |
Ref |
Expression |
1 |
|
psdffval.s |
⊢ 𝑆 = ( 𝐼 mPwSer 𝑅 ) |
2 |
|
psdffval.b |
⊢ 𝐵 = ( Base ‘ 𝑆 ) |
3 |
|
psdffval.d |
⊢ 𝐷 = { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } |
4 |
|
psdffval.i |
⊢ ( 𝜑 → 𝐼 ∈ 𝑉 ) |
5 |
|
psdffval.r |
⊢ ( 𝜑 → 𝑅 ∈ 𝑊 ) |
6 |
|
psdfval.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝐼 ) |
7 |
1 2 3 4 5
|
psdffval |
⊢ ( 𝜑 → ( 𝐼 mPSDer 𝑅 ) = ( 𝑥 ∈ 𝐼 ↦ ( 𝑓 ∈ 𝐵 ↦ ( 𝑘 ∈ 𝐷 ↦ ( ( ( 𝑘 ‘ 𝑥 ) + 1 ) ( .g ‘ 𝑅 ) ( 𝑓 ‘ ( 𝑘 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑥 , 1 , 0 ) ) ) ) ) ) ) ) ) |
8 |
|
fveq2 |
⊢ ( 𝑥 = 𝑋 → ( 𝑘 ‘ 𝑥 ) = ( 𝑘 ‘ 𝑋 ) ) |
9 |
8
|
oveq1d |
⊢ ( 𝑥 = 𝑋 → ( ( 𝑘 ‘ 𝑥 ) + 1 ) = ( ( 𝑘 ‘ 𝑋 ) + 1 ) ) |
10 |
|
eqeq2 |
⊢ ( 𝑥 = 𝑋 → ( 𝑦 = 𝑥 ↔ 𝑦 = 𝑋 ) ) |
11 |
10
|
ifbid |
⊢ ( 𝑥 = 𝑋 → if ( 𝑦 = 𝑥 , 1 , 0 ) = if ( 𝑦 = 𝑋 , 1 , 0 ) ) |
12 |
11
|
mpteq2dv |
⊢ ( 𝑥 = 𝑋 → ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑥 , 1 , 0 ) ) = ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) |
13 |
12
|
oveq2d |
⊢ ( 𝑥 = 𝑋 → ( 𝑘 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑥 , 1 , 0 ) ) ) = ( 𝑘 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) |
14 |
13
|
fveq2d |
⊢ ( 𝑥 = 𝑋 → ( 𝑓 ‘ ( 𝑘 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑥 , 1 , 0 ) ) ) ) = ( 𝑓 ‘ ( 𝑘 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) |
15 |
9 14
|
oveq12d |
⊢ ( 𝑥 = 𝑋 → ( ( ( 𝑘 ‘ 𝑥 ) + 1 ) ( .g ‘ 𝑅 ) ( 𝑓 ‘ ( 𝑘 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑥 , 1 , 0 ) ) ) ) ) = ( ( ( 𝑘 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( 𝑓 ‘ ( 𝑘 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) ) |
16 |
15
|
mpteq2dv |
⊢ ( 𝑥 = 𝑋 → ( 𝑘 ∈ 𝐷 ↦ ( ( ( 𝑘 ‘ 𝑥 ) + 1 ) ( .g ‘ 𝑅 ) ( 𝑓 ‘ ( 𝑘 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑥 , 1 , 0 ) ) ) ) ) ) = ( 𝑘 ∈ 𝐷 ↦ ( ( ( 𝑘 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( 𝑓 ‘ ( 𝑘 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) ) ) |
17 |
16
|
mpteq2dv |
⊢ ( 𝑥 = 𝑋 → ( 𝑓 ∈ 𝐵 ↦ ( 𝑘 ∈ 𝐷 ↦ ( ( ( 𝑘 ‘ 𝑥 ) + 1 ) ( .g ‘ 𝑅 ) ( 𝑓 ‘ ( 𝑘 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑥 , 1 , 0 ) ) ) ) ) ) ) = ( 𝑓 ∈ 𝐵 ↦ ( 𝑘 ∈ 𝐷 ↦ ( ( ( 𝑘 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( 𝑓 ‘ ( 𝑘 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) ) ) ) |
18 |
17
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑥 = 𝑋 ) → ( 𝑓 ∈ 𝐵 ↦ ( 𝑘 ∈ 𝐷 ↦ ( ( ( 𝑘 ‘ 𝑥 ) + 1 ) ( .g ‘ 𝑅 ) ( 𝑓 ‘ ( 𝑘 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑥 , 1 , 0 ) ) ) ) ) ) ) = ( 𝑓 ∈ 𝐵 ↦ ( 𝑘 ∈ 𝐷 ↦ ( ( ( 𝑘 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( 𝑓 ‘ ( 𝑘 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) ) ) ) |
19 |
2
|
fvexi |
⊢ 𝐵 ∈ V |
20 |
19
|
a1i |
⊢ ( 𝜑 → 𝐵 ∈ V ) |
21 |
20
|
mptexd |
⊢ ( 𝜑 → ( 𝑓 ∈ 𝐵 ↦ ( 𝑘 ∈ 𝐷 ↦ ( ( ( 𝑘 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( 𝑓 ‘ ( 𝑘 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) ) ) ∈ V ) |
22 |
7 18 6 21
|
fvmptd |
⊢ ( 𝜑 → ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) = ( 𝑓 ∈ 𝐵 ↦ ( 𝑘 ∈ 𝐷 ↦ ( ( ( 𝑘 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( 𝑓 ‘ ( 𝑘 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) ) ) ) |