Theorem omsval 34437

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 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω)} ↦ Σ*𝑦 ∈ 𝑥(𝑅‘𝑦)), (0[,]+∞), < )))

Distinct variable group:   𝑥,𝑎,𝑦,𝑧,𝑅

Proof of Theorem omsval

Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-oms 34436	. 2 ⊢ toOMeas = (𝑟 ∈ V ↦ (𝑎 ∈ 𝒫 ∪ dom 𝑟 ↦ inf(ran (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑟 ∣ (𝑎 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω)} ↦ Σ*𝑦 ∈ 𝑥(𝑟‘𝑦)), (0[,]+∞), < )))
2		dmeq 5858	. . . . 5 ⊢ (𝑟 = 𝑅 → dom 𝑟 = dom 𝑅)
3	2	unieqd 4863	. . . 4 ⊢ (𝑟 = 𝑅 → ∪ dom 𝑟 = ∪ dom 𝑅)
4	3	pweqd 4558	. . 3 ⊢ (𝑟 = 𝑅 → 𝒫 ∪ dom 𝑟 = 𝒫 ∪ dom 𝑅)
5	2	pweqd 4558	. . . . . . 7 ⊢ (𝑟 = 𝑅 → 𝒫 dom 𝑟 = 𝒫 dom 𝑅)
6		rabeq 3403	. . . . . . 7 ⊢ (𝒫 dom 𝑟 = 𝒫 dom 𝑅 → {𝑧 ∈ 𝒫 dom 𝑟 ∣ (𝑎 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω)} = {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝑎 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω)})
7	5, 6	syl 17	. . . . . 6 ⊢ (𝑟 = 𝑅 → {𝑧 ∈ 𝒫 dom 𝑟 ∣ (𝑎 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω)} = {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝑎 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω)})
8		simpl 482	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑦 ∈ 𝑥) → 𝑟 = 𝑅)
9	8	fveq1d 6842	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑦 ∈ 𝑥) → (𝑟‘𝑦) = (𝑅‘𝑦))
10	9	esumeq2dv 34182	. . . . . 6 ⊢ (𝑟 = 𝑅 → Σ*𝑦 ∈ 𝑥(𝑟‘𝑦) = Σ*𝑦 ∈ 𝑥(𝑅‘𝑦))
11	7, 10	mpteq12dv 5172	. . . . 5 ⊢ (𝑟 = 𝑅 → (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑟 ∣ (𝑎 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω)} ↦ Σ*𝑦 ∈ 𝑥(𝑟‘𝑦)) = (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝑎 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω)} ↦ Σ*𝑦 ∈ 𝑥(𝑅‘𝑦)))
12	11	rneqd 5893	. . . 4 ⊢ (𝑟 = 𝑅 → ran (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑟 ∣ (𝑎 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω)} ↦ Σ*𝑦 ∈ 𝑥(𝑟‘𝑦)) = ran (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝑎 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω)} ↦ Σ*𝑦 ∈ 𝑥(𝑅‘𝑦)))
13	12	infeq1d 9391	. . 3 ⊢ (𝑟 = 𝑅 → inf(ran (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑟 ∣ (𝑎 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω)} ↦ Σ*𝑦 ∈ 𝑥(𝑟‘𝑦)), (0[,]+∞), < ) = inf(ran (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝑎 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω)} ↦ Σ*𝑦 ∈ 𝑥(𝑅‘𝑦)), (0[,]+∞), < ))
14	4, 13	mpteq12dv 5172	. 2 ⊢ (𝑟 = 𝑅 → (𝑎 ∈ 𝒫 ∪ dom 𝑟 ↦ inf(ran (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑟 ∣ (𝑎 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω)} ↦ Σ*𝑦 ∈ 𝑥(𝑟‘𝑦)), (0[,]+∞), < )) = (𝑎 ∈ 𝒫 ∪ dom 𝑅 ↦ inf(ran (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝑎 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω)} ↦ Σ*𝑦 ∈ 𝑥(𝑅‘𝑦)), (0[,]+∞), < )))
15		id 22	. 2 ⊢ (𝑅 ∈ V → 𝑅 ∈ V)
16		dmexg 7852	. . 3 ⊢ (𝑅 ∈ V → dom 𝑅 ∈ V)
17		uniexg 7694	. . 3 ⊢ (dom 𝑅 ∈ V → ∪ dom 𝑅 ∈ V)
18		pwexg 5320	. . 3 ⊢ (∪ dom 𝑅 ∈ V → 𝒫 ∪ dom 𝑅 ∈ V)
19		mptexg 7176	. . 3 ⊢ (𝒫 ∪ dom 𝑅 ∈ V → (𝑎 ∈ 𝒫 ∪ dom 𝑅 ↦ inf(ran (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝑎 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω)} ↦ Σ*𝑦 ∈ 𝑥(𝑅‘𝑦)), (0[,]+∞), < )) ∈ V)
20	16, 17, 18, 19	4syl 19	. 2 ⊢ (𝑅 ∈ V → (𝑎 ∈ 𝒫 ∪ dom 𝑅 ↦ inf(ran (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝑎 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω)} ↦ Σ*𝑦 ∈ 𝑥(𝑅‘𝑦)), (0[,]+∞), < )) ∈ V)
21	1, 14, 15, 20	fvmptd3 6971	1 ⊢ (𝑅 ∈ V → (toOMeas‘𝑅) = (𝑎 ∈ 𝒫 ∪ dom 𝑅 ↦ inf(ran (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝑎 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω)} ↦ Σ*𝑦 ∈ 𝑥(𝑅‘𝑦)), (0[,]+∞), < )))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {crab 3389  Vcvv 3429   ⊆ wss 3889  𝒫 cpw 4541  ∪ cuni 4850   class class class wbr 5085   ↦ cmpt 5166  dom cdm 5631  ran crn 5632  ‘cfv 6498  (class class class)co 7367  ωcom 7817   ≼ cdom 8891  infcinf 9354  0cc0 11038  +∞cpnf 11176   < clt 11179  [,]cicc 13301  Σ^*cesum 34171  toOMeascoms 34435

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-sup 9355  df-inf 9356  df-esum 34172  df-oms 34436

This theorem is referenced by:  omsfval  34438  omsf  34440