Metamath Proof Explorer


Theorem omsval

Description: Value of the function mapping a content function to the corresponding outer measure. (Contributed by Thierry Arnoux, 15-Sep-2019) (Revised by AV, 4-Oct-2020)

Ref Expression
Assertion omsval ( 𝑅 ∈ V → ( toOMeas ‘ 𝑅 ) = ( 𝑎 ∈ 𝒫 dom 𝑅 ↦ inf ( ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝑎 𝑧𝑧 ≼ ω ) } ↦ Σ* 𝑦𝑥 ( 𝑅𝑦 ) ) , ( 0 [,] +∞ ) , < ) ) )

Proof

Step Hyp Ref Expression
1 df-oms toOMeas = ( 𝑟 ∈ V ↦ ( 𝑎 ∈ 𝒫 dom 𝑟 ↦ inf ( ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑟 ∣ ( 𝑎 𝑧𝑧 ≼ ω ) } ↦ Σ* 𝑦𝑥 ( 𝑟𝑦 ) ) , ( 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 𝑧𝑧 ≼ ω ) } = { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝑎 𝑧𝑧 ≼ ω ) } )
7 5 6 syl ( 𝑟 = 𝑅 → { 𝑧 ∈ 𝒫 dom 𝑟 ∣ ( 𝑎 𝑧𝑧 ≼ ω ) } = { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝑎 𝑧𝑧 ≼ ω ) } )
8 simpl ( ( 𝑟 = 𝑅𝑦𝑥 ) → 𝑟 = 𝑅 )
9 8 fveq1d ( ( 𝑟 = 𝑅𝑦𝑥 ) → ( 𝑟𝑦 ) = ( 𝑅𝑦 ) )
10 9 esumeq2dv ( 𝑟 = 𝑅 → Σ* 𝑦𝑥 ( 𝑟𝑦 ) = Σ* 𝑦𝑥 ( 𝑅𝑦 ) )
11 7 10 mpteq12dv ( 𝑟 = 𝑅 → ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑟 ∣ ( 𝑎 𝑧𝑧 ≼ ω ) } ↦ Σ* 𝑦𝑥 ( 𝑟𝑦 ) ) = ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝑎 𝑧𝑧 ≼ ω ) } ↦ Σ* 𝑦𝑥 ( 𝑅𝑦 ) ) )
12 11 rneqd ( 𝑟 = 𝑅 → ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑟 ∣ ( 𝑎 𝑧𝑧 ≼ ω ) } ↦ Σ* 𝑦𝑥 ( 𝑟𝑦 ) ) = ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝑎 𝑧𝑧 ≼ ω ) } ↦ Σ* 𝑦𝑥 ( 𝑅𝑦 ) ) )
13 12 infeq1d ( 𝑟 = 𝑅 → inf ( ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑟 ∣ ( 𝑎 𝑧𝑧 ≼ ω ) } ↦ Σ* 𝑦𝑥 ( 𝑟𝑦 ) ) , ( 0 [,] +∞ ) , < ) = inf ( ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝑎 𝑧𝑧 ≼ ω ) } ↦ Σ* 𝑦𝑥 ( 𝑅𝑦 ) ) , ( 0 [,] +∞ ) , < ) )
14 4 13 mpteq12dv ( 𝑟 = 𝑅 → ( 𝑎 ∈ 𝒫 dom 𝑟 ↦ inf ( ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑟 ∣ ( 𝑎 𝑧𝑧 ≼ ω ) } ↦ Σ* 𝑦𝑥 ( 𝑟𝑦 ) ) , ( 0 [,] +∞ ) , < ) ) = ( 𝑎 ∈ 𝒫 dom 𝑅 ↦ inf ( ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝑎 𝑧𝑧 ≼ ω ) } ↦ Σ* 𝑦𝑥 ( 𝑅𝑦 ) ) , ( 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 𝑧𝑧 ≼ ω ) } ↦ Σ* 𝑦𝑥 ( 𝑅𝑦 ) ) , ( 0 [,] +∞ ) , < ) ) ∈ V )
20 16 17 18 19 4syl ( 𝑅 ∈ V → ( 𝑎 ∈ 𝒫 dom 𝑅 ↦ inf ( ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝑎 𝑧𝑧 ≼ ω ) } ↦ Σ* 𝑦𝑥 ( 𝑅𝑦 ) ) , ( 0 [,] +∞ ) , < ) ) ∈ V )
21 1 14 15 20 fvmptd3 ( 𝑅 ∈ V → ( toOMeas ‘ 𝑅 ) = ( 𝑎 ∈ 𝒫 dom 𝑅 ↦ inf ( ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝑎 𝑧𝑧 ≼ ω ) } ↦ Σ* 𝑦𝑥 ( 𝑅𝑦 ) ) , ( 0 [,] +∞ ) , < ) ) )