Theorem omsval 34590

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 34589	. 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 5879	. . . . 5 ⊢ (𝑟 = 𝑅 → dom 𝑟 = dom 𝑅)
3	2	unieqd 4878	. . . 4 ⊢ (𝑟 = 𝑅 → ∪ dom 𝑟 = ∪ dom 𝑅)
4	3	pweqd 4572	. . 3 ⊢ (𝑟 = 𝑅 → 𝒫 ∪ dom 𝑟 = 𝒫 ∪ dom 𝑅)
5	2	pweqd 4572	. . . . . . 7 ⊢ (𝑟 = 𝑅 → 𝒫 dom 𝑟 = 𝒫 dom 𝑅)
6		rabeq 3428	. . . . . . 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 486	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑦 ∈ 𝑥) → 𝑟 = 𝑅)
9	8	fveq1d 6869	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑦 ∈ 𝑥) → (𝑟‘𝑦) = (𝑅‘𝑦))
10	9	esumeq2dv 34335	. . . . . 6 ⊢ (𝑟 = 𝑅 → Σ*𝑦 ∈ 𝑥(𝑟‘𝑦) = Σ*𝑦 ∈ 𝑥(𝑅‘𝑦))
11	7, 10	mpteq12dv 5187	. . . . 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 5914	. . . 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 9424	. . 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 5187	. 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 7882	. . 3 ⊢ (𝑅 ∈ V → dom 𝑅 ∈ V)
17		uniexg 7723	. . 3 ⊢ (dom 𝑅 ∈ V → ∪ dom 𝑅 ∈ V)
18		pwexg 5335	. . 3 ⊢ (∪ dom 𝑅 ∈ V → 𝒫 ∪ dom 𝑅 ∈ V)
19		mptexg 7205	. . 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 6999	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 399   = wceq 1560   ∈ wcel 2142  {crab 3414  Vcvv 3454   ⊆ wss 3904  𝒫 cpw 4555  ∪ cuni 4865   class class class wbr 5100   ↦ cmpt 5181  dom cdm 5647  ran crn 5648  ‘cfv 6521  (class class class)co 7396  ωcom 7846   ≼ cdom 8925  infcinf 9387  0cc0 11073  +∞cpnf 11213   < clt 11216  [,]cicc 13352  Σ^*cesum 34324  toOMeascoms 34588

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-ov 7399  df-sup 9388  df-inf 9389  df-esum 34325  df-oms 34589

This theorem is referenced by:  omsfval  34591  omsf  34593