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Theorem omsval 31450
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.)
Assertion
Ref Expression
omsval (𝑅 ∈ V → (toOMeas‘𝑅) = (𝑎 ∈ 𝒫 dom 𝑅 ↦ inf(ran (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝑎 𝑧𝑧 ≼ ω)} ↦ Σ*𝑦𝑥(𝑅𝑦)), (0[,]+∞), < )))
Distinct variable group:   𝑥,𝑎,𝑦,𝑧,𝑅

Proof of Theorem omsval
Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 df-oms 31449 . 2 toOMeas = (𝑟 ∈ V ↦ (𝑎 ∈ 𝒫 dom 𝑟 ↦ inf(ran (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑟 ∣ (𝑎 𝑧𝑧 ≼ ω)} ↦ Σ*𝑦𝑥(𝑟𝑦)), (0[,]+∞), < )))
2 dmeq 5765 . . . . 5 (𝑟 = 𝑅 → dom 𝑟 = dom 𝑅)
32unieqd 4840 . . . 4 (𝑟 = 𝑅 dom 𝑟 = dom 𝑅)
43pweqd 4540 . . 3 (𝑟 = 𝑅 → 𝒫 dom 𝑟 = 𝒫 dom 𝑅)
52pweqd 4540 . . . . . . 7 (𝑟 = 𝑅 → 𝒫 dom 𝑟 = 𝒫 dom 𝑅)
6 rabeq 3481 . . . . . . 7 (𝒫 dom 𝑟 = 𝒫 dom 𝑅 → {𝑧 ∈ 𝒫 dom 𝑟 ∣ (𝑎 𝑧𝑧 ≼ ω)} = {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝑎 𝑧𝑧 ≼ ω)})
75, 6syl 17 . . . . . 6 (𝑟 = 𝑅 → {𝑧 ∈ 𝒫 dom 𝑟 ∣ (𝑎 𝑧𝑧 ≼ ω)} = {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝑎 𝑧𝑧 ≼ ω)})
8 simpl 483 . . . . . . . 8 ((𝑟 = 𝑅𝑦𝑥) → 𝑟 = 𝑅)
98fveq1d 6665 . . . . . . 7 ((𝑟 = 𝑅𝑦𝑥) → (𝑟𝑦) = (𝑅𝑦))
109esumeq2dv 31196 . . . . . 6 (𝑟 = 𝑅 → Σ*𝑦𝑥(𝑟𝑦) = Σ*𝑦𝑥(𝑅𝑦))
117, 10mpteq12dv 5142 . . . . 5 (𝑟 = 𝑅 → (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑟 ∣ (𝑎 𝑧𝑧 ≼ ω)} ↦ Σ*𝑦𝑥(𝑟𝑦)) = (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝑎 𝑧𝑧 ≼ ω)} ↦ Σ*𝑦𝑥(𝑅𝑦)))
1211rneqd 5801 . . . 4 (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)))
1312infeq1d 8929 . . 3 (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[,]+∞), < ))
144, 13mpteq12dv 5142 . 2 (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 22 . 2 (𝑅 ∈ V → 𝑅 ∈ V)
16 dmexg 7602 . . 3 (𝑅 ∈ V → dom 𝑅 ∈ V)
17 uniexg 7456 . . 3 (dom 𝑅 ∈ V → dom 𝑅 ∈ V)
18 pwexg 5270 . . 3 ( dom 𝑅 ∈ V → 𝒫 dom 𝑅 ∈ V)
19 mptexg 6975 . . 3 (𝒫 dom 𝑅 ∈ V → (𝑎 ∈ 𝒫 dom 𝑅 ↦ inf(ran (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝑎 𝑧𝑧 ≼ ω)} ↦ Σ*𝑦𝑥(𝑅𝑦)), (0[,]+∞), < )) ∈ V)
2016, 17, 18, 194syl 19 . 2 (𝑅 ∈ V → (𝑎 ∈ 𝒫 dom 𝑅 ↦ inf(ran (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝑎 𝑧𝑧 ≼ ω)} ↦ Σ*𝑦𝑥(𝑅𝑦)), (0[,]+∞), < )) ∈ V)
211, 14, 15, 20fvmptd3 6783 1 (𝑅 ∈ V → (toOMeas‘𝑅) = (𝑎 ∈ 𝒫 dom 𝑅 ↦ inf(ran (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝑎 𝑧𝑧 ≼ ω)} ↦ Σ*𝑦𝑥(𝑅𝑦)), (0[,]+∞), < )))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  {crab 3139  Vcvv 3492  wss 3933  𝒫 cpw 4535   cuni 4830   class class class wbr 5057  cmpt 5137  dom cdm 5548  ran crn 5549  cfv 6348  (class class class)co 7145  ωcom 7569  cdom 8495  infcinf 8893  0cc0 10525  +∞cpnf 10660   < clt 10663  [,]cicc 12729  Σ*cesum 31185  toOMeascoms 31448
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-sup 8894  df-inf 8895  df-esum 31186  df-oms 31449
This theorem is referenced by:  omsfval  31451  omsf  31453
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