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Theorem omsval 33292
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 33291 . 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 5904 . . . . 5 (π‘Ÿ = 𝑅 β†’ dom π‘Ÿ = dom 𝑅)
32unieqd 4923 . . . 4 (π‘Ÿ = 𝑅 β†’ βˆͺ dom π‘Ÿ = βˆͺ dom 𝑅)
43pweqd 4620 . . 3 (π‘Ÿ = 𝑅 β†’ 𝒫 βˆͺ dom π‘Ÿ = 𝒫 βˆͺ dom 𝑅)
52pweqd 4620 . . . . . . 7 (π‘Ÿ = 𝑅 β†’ 𝒫 dom π‘Ÿ = 𝒫 dom 𝑅)
6 rabeq 3447 . . . . . . 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 484 . . . . . . . 8 ((π‘Ÿ = 𝑅 ∧ 𝑦 ∈ π‘₯) β†’ π‘Ÿ = 𝑅)
98fveq1d 6894 . . . . . . 7 ((π‘Ÿ = 𝑅 ∧ 𝑦 ∈ π‘₯) β†’ (π‘Ÿβ€˜π‘¦) = (π‘…β€˜π‘¦))
109esumeq2dv 33036 . . . . . 6 (π‘Ÿ = 𝑅 β†’ Ξ£*𝑦 ∈ π‘₯(π‘Ÿβ€˜π‘¦) = Ξ£*𝑦 ∈ π‘₯(π‘…β€˜π‘¦))
117, 10mpteq12dv 5240 . . . . 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 5938 . . . 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 9472 . . 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 5240 . 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 7894 . . 3 (𝑅 ∈ V β†’ dom 𝑅 ∈ V)
17 uniexg 7730 . . 3 (dom 𝑅 ∈ V β†’ βˆͺ dom 𝑅 ∈ V)
18 pwexg 5377 . . 3 (βˆͺ dom 𝑅 ∈ V β†’ 𝒫 βˆͺ dom 𝑅 ∈ V)
19 mptexg 7223 . . 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 7022 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 397   = wceq 1542   ∈ wcel 2107  {crab 3433  Vcvv 3475   βŠ† wss 3949  π’« cpw 4603  βˆͺ cuni 4909   class class class wbr 5149   ↦ cmpt 5232  dom cdm 5677  ran crn 5678  β€˜cfv 6544  (class class class)co 7409  Ο‰com 7855   β‰Ό cdom 8937  infcinf 9436  0cc0 11110  +∞cpnf 11245   < clt 11248  [,]cicc 13327  Ξ£*cesum 33025  toOMeascoms 33290
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-sup 9437  df-inf 9438  df-esum 33026  df-oms 33291
This theorem is referenced by:  omsfval  33293  omsf  33295
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