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Theorem omsval 30685
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 30684 . . 3 toOMeas = (𝑟 ∈ V ↦ (𝑎 ∈ 𝒫 dom 𝑟 ↦ inf(ran (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑟 ∣ (𝑎 𝑧𝑧 ≼ ω)} ↦ Σ*𝑦𝑥(𝑟𝑦)), (0[,]+∞), < )))
21a1i 11 . 2 (𝑅 ∈ V → toOMeas = (𝑟 ∈ V ↦ (𝑎 ∈ 𝒫 dom 𝑟 ↦ inf(ran (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑟 ∣ (𝑎 𝑧𝑧 ≼ ω)} ↦ Σ*𝑦𝑥(𝑟𝑦)), (0[,]+∞), < ))))
3 dmeq 5479 . . . . . 6 (𝑟 = 𝑅 → dom 𝑟 = dom 𝑅)
43unieqd 4598 . . . . 5 (𝑟 = 𝑅 dom 𝑟 = dom 𝑅)
54pweqd 4307 . . . 4 (𝑟 = 𝑅 → 𝒫 dom 𝑟 = 𝒫 dom 𝑅)
63pweqd 4307 . . . . . . . 8 (𝑟 = 𝑅 → 𝒫 dom 𝑟 = 𝒫 dom 𝑅)
7 rabeq 3332 . . . . . . . 8 (𝒫 dom 𝑟 = 𝒫 dom 𝑅 → {𝑧 ∈ 𝒫 dom 𝑟 ∣ (𝑎 𝑧𝑧 ≼ ω)} = {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝑎 𝑧𝑧 ≼ ω)})
86, 7syl 17 . . . . . . 7 (𝑟 = 𝑅 → {𝑧 ∈ 𝒫 dom 𝑟 ∣ (𝑎 𝑧𝑧 ≼ ω)} = {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝑎 𝑧𝑧 ≼ ω)})
9 simpl 474 . . . . . . . . 9 ((𝑟 = 𝑅𝑦𝑥) → 𝑟 = 𝑅)
109fveq1d 6355 . . . . . . . 8 ((𝑟 = 𝑅𝑦𝑥) → (𝑟𝑦) = (𝑅𝑦))
1110esumeq2dv 30430 . . . . . . 7 (𝑟 = 𝑅 → Σ*𝑦𝑥(𝑟𝑦) = Σ*𝑦𝑥(𝑅𝑦))
128, 11mpteq12dv 4885 . . . . . 6 (𝑟 = 𝑅 → (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑟 ∣ (𝑎 𝑧𝑧 ≼ ω)} ↦ Σ*𝑦𝑥(𝑟𝑦)) = (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝑎 𝑧𝑧 ≼ ω)} ↦ Σ*𝑦𝑥(𝑅𝑦)))
1312rneqd 5508 . . . . 5 (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)))
1413infeq1d 8550 . . . 4 (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[,]+∞), < ))
155, 14mpteq12dv 4885 . . 3 (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[,]+∞), < )))
1615adantl 473 . 2 ((𝑅 ∈ V ∧ 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[,]+∞), < )))
17 id 22 . 2 (𝑅 ∈ V → 𝑅 ∈ V)
18 dmexg 7263 . . 3 (𝑅 ∈ V → dom 𝑅 ∈ V)
19 uniexg 7121 . . 3 (dom 𝑅 ∈ V → dom 𝑅 ∈ V)
20 pwexg 4999 . . 3 ( dom 𝑅 ∈ V → 𝒫 dom 𝑅 ∈ V)
21 mptexg 6649 . . 3 (𝒫 dom 𝑅 ∈ V → (𝑎 ∈ 𝒫 dom 𝑅 ↦ inf(ran (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝑎 𝑧𝑧 ≼ ω)} ↦ Σ*𝑦𝑥(𝑅𝑦)), (0[,]+∞), < )) ∈ V)
2218, 19, 20, 214syl 19 . 2 (𝑅 ∈ V → (𝑎 ∈ 𝒫 dom 𝑅 ↦ inf(ran (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝑎 𝑧𝑧 ≼ ω)} ↦ Σ*𝑦𝑥(𝑅𝑦)), (0[,]+∞), < )) ∈ V)
232, 16, 17, 22fvmptd 6451 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 383   = wceq 1632  wcel 2139  {crab 3054  Vcvv 3340  wss 3715  𝒫 cpw 4302   cuni 4588   class class class wbr 4804  cmpt 4881  dom cdm 5266  ran crn 5267  cfv 6049  (class class class)co 6814  ωcom 7231  cdom 8121  infcinf 8514  0cc0 10148  +∞cpnf 10283   < clt 10286  [,]cicc 12391  Σ*cesum 30419  toOMeascoms 30683
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6817  df-sup 8515  df-inf 8516  df-esum 30420  df-oms 30684
This theorem is referenced by:  omsfval  30686  omsf  30688
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