Step |
Hyp |
Ref |
Expression |
1 |
|
df-oms |
⊢ toOMeas = ( 𝑟 ∈ V ↦ ( 𝑎 ∈ 𝒫 ∪ dom 𝑟 ↦ inf ( ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑟 ∣ ( 𝑎 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑦 ∈ 𝑥 ( 𝑟 ‘ 𝑦 ) ) , ( 0 [,] +∞ ) , < ) ) ) |
2 |
|
dmeq |
⊢ ( 𝑟 = 𝑅 → dom 𝑟 = dom 𝑅 ) |
3 |
2
|
unieqd |
⊢ ( 𝑟 = 𝑅 → ∪ dom 𝑟 = ∪ dom 𝑅 ) |
4 |
3
|
pweqd |
⊢ ( 𝑟 = 𝑅 → 𝒫 ∪ dom 𝑟 = 𝒫 ∪ dom 𝑅 ) |
5 |
2
|
pweqd |
⊢ ( 𝑟 = 𝑅 → 𝒫 dom 𝑟 = 𝒫 dom 𝑅 ) |
6 |
|
rabeq |
⊢ ( 𝒫 dom 𝑟 = 𝒫 dom 𝑅 → { 𝑧 ∈ 𝒫 dom 𝑟 ∣ ( 𝑎 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } = { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝑎 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ) |
7 |
5 6
|
syl |
⊢ ( 𝑟 = 𝑅 → { 𝑧 ∈ 𝒫 dom 𝑟 ∣ ( 𝑎 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } = { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝑎 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ) |
8 |
|
simpl |
⊢ ( ( 𝑟 = 𝑅 ∧ 𝑦 ∈ 𝑥 ) → 𝑟 = 𝑅 ) |
9 |
8
|
fveq1d |
⊢ ( ( 𝑟 = 𝑅 ∧ 𝑦 ∈ 𝑥 ) → ( 𝑟 ‘ 𝑦 ) = ( 𝑅 ‘ 𝑦 ) ) |
10 |
9
|
esumeq2dv |
⊢ ( 𝑟 = 𝑅 → Σ* 𝑦 ∈ 𝑥 ( 𝑟 ‘ 𝑦 ) = Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ) |
11 |
7 10
|
mpteq12dv |
⊢ ( 𝑟 = 𝑅 → ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑟 ∣ ( 𝑎 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑦 ∈ 𝑥 ( 𝑟 ‘ 𝑦 ) ) = ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝑎 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ) ) |
12 |
11
|
rneqd |
⊢ ( 𝑟 = 𝑅 → ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑟 ∣ ( 𝑎 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑦 ∈ 𝑥 ( 𝑟 ‘ 𝑦 ) ) = ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝑎 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ) ) |
13 |
12
|
infeq1d |
⊢ ( 𝑟 = 𝑅 → inf ( ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑟 ∣ ( 𝑎 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑦 ∈ 𝑥 ( 𝑟 ‘ 𝑦 ) ) , ( 0 [,] +∞ ) , < ) = inf ( ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝑎 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ) , ( 0 [,] +∞ ) , < ) ) |
14 |
4 13
|
mpteq12dv |
⊢ ( 𝑟 = 𝑅 → ( 𝑎 ∈ 𝒫 ∪ dom 𝑟 ↦ inf ( ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑟 ∣ ( 𝑎 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑦 ∈ 𝑥 ( 𝑟 ‘ 𝑦 ) ) , ( 0 [,] +∞ ) , < ) ) = ( 𝑎 ∈ 𝒫 ∪ dom 𝑅 ↦ inf ( ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝑎 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ) , ( 0 [,] +∞ ) , < ) ) ) |
15 |
|
id |
⊢ ( 𝑅 ∈ V → 𝑅 ∈ V ) |
16 |
|
dmexg |
⊢ ( 𝑅 ∈ V → dom 𝑅 ∈ V ) |
17 |
|
uniexg |
⊢ ( dom 𝑅 ∈ V → ∪ dom 𝑅 ∈ V ) |
18 |
|
pwexg |
⊢ ( ∪ dom 𝑅 ∈ V → 𝒫 ∪ dom 𝑅 ∈ V ) |
19 |
|
mptexg |
⊢ ( 𝒫 ∪ dom 𝑅 ∈ V → ( 𝑎 ∈ 𝒫 ∪ dom 𝑅 ↦ inf ( ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝑎 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ) , ( 0 [,] +∞ ) , < ) ) ∈ V ) |
20 |
16 17 18 19
|
4syl |
⊢ ( 𝑅 ∈ V → ( 𝑎 ∈ 𝒫 ∪ dom 𝑅 ↦ inf ( ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝑎 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ) , ( 0 [,] +∞ ) , < ) ) ∈ V ) |
21 |
1 14 15 20
|
fvmptd3 |
⊢ ( 𝑅 ∈ V → ( toOMeas ‘ 𝑅 ) = ( 𝑎 ∈ 𝒫 ∪ dom 𝑅 ↦ inf ( ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝑎 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ) , ( 0 [,] +∞ ) , < ) ) ) |