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Theorem omsval 33590
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 β†’ (toOMeasβ€˜π‘…) = (π‘Ž ∈ 𝒫 βˆͺ dom 𝑅 ↦ inf(ran (π‘₯ ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (π‘Ž βŠ† βˆͺ 𝑧 ∧ 𝑧 β‰Ό Ο‰)} ↦ Ξ£*𝑦 ∈ π‘₯(π‘…β€˜π‘¦)), (0[,]+∞), < )))
Distinct variable group:   π‘₯,π‘Ž,𝑦,𝑧,𝑅

Proof of Theorem omsval
Dummy variable π‘Ÿ is distinct from all other variables.
StepHypRef Expression
1 df-oms 33589 . 2 toOMeas = (π‘Ÿ ∈ V ↦ (π‘Ž ∈ 𝒫 βˆͺ dom π‘Ÿ ↦ inf(ran (π‘₯ ∈ {𝑧 ∈ 𝒫 dom π‘Ÿ ∣ (π‘Ž βŠ† βˆͺ 𝑧 ∧ 𝑧 β‰Ό Ο‰)} ↦ Ξ£*𝑦 ∈ π‘₯(π‘Ÿβ€˜π‘¦)), (0[,]+∞), < )))
2 dmeq 5902 . . . . 5 (π‘Ÿ = 𝑅 β†’ dom π‘Ÿ = dom 𝑅)
32unieqd 4921 . . . 4 (π‘Ÿ = 𝑅 β†’ βˆͺ dom π‘Ÿ = βˆͺ dom 𝑅)
43pweqd 4618 . . 3 (π‘Ÿ = 𝑅 β†’ 𝒫 βˆͺ dom π‘Ÿ = 𝒫 βˆͺ dom 𝑅)
52pweqd 4618 . . . . . . 7 (π‘Ÿ = 𝑅 β†’ 𝒫 dom π‘Ÿ = 𝒫 dom 𝑅)
6 rabeq 3444 . . . . . . 7 (𝒫 dom π‘Ÿ = 𝒫 dom 𝑅 β†’ {𝑧 ∈ 𝒫 dom π‘Ÿ ∣ (π‘Ž βŠ† βˆͺ 𝑧 ∧ 𝑧 β‰Ό Ο‰)} = {𝑧 ∈ 𝒫 dom 𝑅 ∣ (π‘Ž βŠ† βˆͺ 𝑧 ∧ 𝑧 β‰Ό Ο‰)})
75, 6syl 17 . . . . . 6 (π‘Ÿ = 𝑅 β†’ {𝑧 ∈ 𝒫 dom π‘Ÿ ∣ (π‘Ž βŠ† βˆͺ 𝑧 ∧ 𝑧 β‰Ό Ο‰)} = {𝑧 ∈ 𝒫 dom 𝑅 ∣ (π‘Ž βŠ† βˆͺ 𝑧 ∧ 𝑧 β‰Ό Ο‰)})
8 simpl 481 . . . . . . . 8 ((π‘Ÿ = 𝑅 ∧ 𝑦 ∈ π‘₯) β†’ π‘Ÿ = 𝑅)
98fveq1d 6892 . . . . . . 7 ((π‘Ÿ = 𝑅 ∧ 𝑦 ∈ π‘₯) β†’ (π‘Ÿβ€˜π‘¦) = (π‘…β€˜π‘¦))
109esumeq2dv 33334 . . . . . 6 (π‘Ÿ = 𝑅 β†’ Ξ£*𝑦 ∈ π‘₯(π‘Ÿβ€˜π‘¦) = Ξ£*𝑦 ∈ π‘₯(π‘…β€˜π‘¦))
117, 10mpteq12dv 5238 . . . . 5 (π‘Ÿ = 𝑅 β†’ (π‘₯ ∈ {𝑧 ∈ 𝒫 dom π‘Ÿ ∣ (π‘Ž βŠ† βˆͺ 𝑧 ∧ 𝑧 β‰Ό Ο‰)} ↦ Ξ£*𝑦 ∈ π‘₯(π‘Ÿβ€˜π‘¦)) = (π‘₯ ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (π‘Ž βŠ† βˆͺ 𝑧 ∧ 𝑧 β‰Ό Ο‰)} ↦ Ξ£*𝑦 ∈ π‘₯(π‘…β€˜π‘¦)))
1211rneqd 5936 . . . 4 (π‘Ÿ = 𝑅 β†’ ran (π‘₯ ∈ {𝑧 ∈ 𝒫 dom π‘Ÿ ∣ (π‘Ž βŠ† βˆͺ 𝑧 ∧ 𝑧 β‰Ό Ο‰)} ↦ Ξ£*𝑦 ∈ π‘₯(π‘Ÿβ€˜π‘¦)) = ran (π‘₯ ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (π‘Ž βŠ† βˆͺ 𝑧 ∧ 𝑧 β‰Ό Ο‰)} ↦ Ξ£*𝑦 ∈ π‘₯(π‘…β€˜π‘¦)))
1312infeq1d 9474 . . 3 (π‘Ÿ = 𝑅 β†’ inf(ran (π‘₯ ∈ {𝑧 ∈ 𝒫 dom π‘Ÿ ∣ (π‘Ž βŠ† βˆͺ 𝑧 ∧ 𝑧 β‰Ό Ο‰)} ↦ Ξ£*𝑦 ∈ π‘₯(π‘Ÿβ€˜π‘¦)), (0[,]+∞), < ) = inf(ran (π‘₯ ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (π‘Ž βŠ† βˆͺ 𝑧 ∧ 𝑧 β‰Ό Ο‰)} ↦ Ξ£*𝑦 ∈ π‘₯(π‘…β€˜π‘¦)), (0[,]+∞), < ))
144, 13mpteq12dv 5238 . 2 (π‘Ÿ = 𝑅 β†’ (π‘Ž ∈ 𝒫 βˆͺ dom π‘Ÿ ↦ inf(ran (π‘₯ ∈ {𝑧 ∈ 𝒫 dom π‘Ÿ ∣ (π‘Ž βŠ† βˆͺ 𝑧 ∧ 𝑧 β‰Ό Ο‰)} ↦ Ξ£*𝑦 ∈ π‘₯(π‘Ÿβ€˜π‘¦)), (0[,]+∞), < )) = (π‘Ž ∈ 𝒫 βˆͺ dom 𝑅 ↦ inf(ran (π‘₯ ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (π‘Ž βŠ† βˆͺ 𝑧 ∧ 𝑧 β‰Ό Ο‰)} ↦ Ξ£*𝑦 ∈ π‘₯(π‘…β€˜π‘¦)), (0[,]+∞), < )))
15 id 22 . 2 (𝑅 ∈ V β†’ 𝑅 ∈ V)
16 dmexg 7896 . . 3 (𝑅 ∈ V β†’ dom 𝑅 ∈ V)
17 uniexg 7732 . . 3 (dom 𝑅 ∈ V β†’ βˆͺ dom 𝑅 ∈ V)
18 pwexg 5375 . . 3 (βˆͺ dom 𝑅 ∈ V β†’ 𝒫 βˆͺ dom 𝑅 ∈ V)
19 mptexg 7224 . . 3 (𝒫 βˆͺ dom 𝑅 ∈ V β†’ (π‘Ž ∈ 𝒫 βˆͺ dom 𝑅 ↦ inf(ran (π‘₯ ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (π‘Ž βŠ† βˆͺ 𝑧 ∧ 𝑧 β‰Ό Ο‰)} ↦ Ξ£*𝑦 ∈ π‘₯(π‘…β€˜π‘¦)), (0[,]+∞), < )) ∈ V)
2016, 17, 18, 194syl 19 . 2 (𝑅 ∈ V β†’ (π‘Ž ∈ 𝒫 βˆͺ dom 𝑅 ↦ inf(ran (π‘₯ ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (π‘Ž βŠ† βˆͺ 𝑧 ∧ 𝑧 β‰Ό Ο‰)} ↦ Ξ£*𝑦 ∈ π‘₯(π‘…β€˜π‘¦)), (0[,]+∞), < )) ∈ V)
211, 14, 15, 20fvmptd3 7020 1 (𝑅 ∈ V β†’ (toOMeasβ€˜π‘…) = (π‘Ž ∈ 𝒫 βˆͺ dom 𝑅 ↦ inf(ran (π‘₯ ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (π‘Ž βŠ† βˆͺ 𝑧 ∧ 𝑧 β‰Ό Ο‰)} ↦ Ξ£*𝑦 ∈ π‘₯(π‘…β€˜π‘¦)), (0[,]+∞), < )))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   = wceq 1539   ∈ wcel 2104  {crab 3430  Vcvv 3472   βŠ† wss 3947  π’« cpw 4601  βˆͺ cuni 4907   class class class wbr 5147   ↦ cmpt 5230  dom cdm 5675  ran crn 5676  β€˜cfv 6542  (class class class)co 7411  Ο‰com 7857   β‰Ό cdom 8939  infcinf 9438  0cc0 11112  +∞cpnf 11249   < clt 11252  [,]cicc 13331  Ξ£*cesum 33323  toOMeascoms 33588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7414  df-sup 9439  df-inf 9440  df-esum 33324  df-oms 33589
This theorem is referenced by:  omsfval  33591  omsf  33593
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