Theorem omsval 33913

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 33912	. 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 5906	. . . . 5 ⊢ (𝑟 = 𝑅 → dom 𝑟 = dom 𝑅)
3	2	unieqd 4921	. . . 4 ⊢ (𝑟 = 𝑅 → ∪ dom 𝑟 = ∪ dom 𝑅)
4	3	pweqd 4620	. . 3 ⊢ (𝑟 = 𝑅 → 𝒫 ∪ dom 𝑟 = 𝒫 ∪ dom 𝑅)
5	2	pweqd 4620	. . . . . . 7 ⊢ (𝑟 = 𝑅 → 𝒫 dom 𝑟 = 𝒫 dom 𝑅)
6		rabeq 3443	. . . . . . 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 6899	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑦 ∈ 𝑥) → (𝑟‘𝑦) = (𝑅‘𝑦))
10	9	esumeq2dv 33657	. . . . . 6 ⊢ (𝑟 = 𝑅 → Σ*𝑦 ∈ 𝑥(𝑟‘𝑦) = Σ*𝑦 ∈ 𝑥(𝑅‘𝑦))
11	7, 10	mpteq12dv 5239	. . . . 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 5940	. . . 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 9501	. . 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 5239	. 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 7909	. . 3 ⊢ (𝑅 ∈ V → dom 𝑅 ∈ V)
17		uniexg 7745	. . 3 ⊢ (dom 𝑅 ∈ V → ∪ dom 𝑅 ∈ V)
18		pwexg 5378	. . 3 ⊢ (∪ dom 𝑅 ∈ V → 𝒫 ∪ dom 𝑅 ∈ V)
19		mptexg 7233	. . 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 7028	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 1534   ∈ wcel 2099  {crab 3429  Vcvv 3471   ⊆ wss 3947  𝒫 cpw 4603  ∪ cuni 4908   class class class wbr 5148   ↦ cmpt 5231  dom cdm 5678  ran crn 5679  ‘cfv 6548  (class class class)co 7420  ωcom 7870   ≼ cdom 8962  infcinf 9465  0cc0 11139  +∞cpnf 11276   < clt 11279  [,]cicc 13360  Σ^*cesum 33646  toOMeascoms 33911

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-ov 7423  df-sup 9466  df-inf 9467  df-esum 33647  df-oms 33912

This theorem is referenced by:  omsfval  33914  omsf  33916