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Theorem caragenelss 46516
Description: An element of the Caratheodory's construction is a subset of the base set of the outer measure. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypotheses
Ref Expression
caragenelss.o (𝜑𝑂 ∈ OutMeas)
caragenelss.s 𝑆 = (CaraGen‘𝑂)
caragenelss.a (𝜑𝐴𝑆)
caragenelss.x 𝑋 = dom 𝑂
Assertion
Ref Expression
caragenelss (𝜑𝐴𝑋)

Proof of Theorem caragenelss
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 caragenelss.a . . . . 5 (𝜑𝐴𝑆)
2 caragenelss.o . . . . . 6 (𝜑𝑂 ∈ OutMeas)
3 caragenelss.s . . . . . 6 𝑆 = (CaraGen‘𝑂)
42, 3caragenel 46510 . . . . 5 (𝜑 → (𝐴𝑆 ↔ (𝐴 ∈ 𝒫 dom 𝑂 ∧ ∀𝑥 ∈ 𝒫 dom 𝑂((𝑂‘(𝑥𝐴)) +𝑒 (𝑂‘(𝑥𝐴))) = (𝑂𝑥))))
51, 4mpbid 232 . . . 4 (𝜑 → (𝐴 ∈ 𝒫 dom 𝑂 ∧ ∀𝑥 ∈ 𝒫 dom 𝑂((𝑂‘(𝑥𝐴)) +𝑒 (𝑂‘(𝑥𝐴))) = (𝑂𝑥)))
65simpld 494 . . 3 (𝜑𝐴 ∈ 𝒫 dom 𝑂)
7 caragenelss.x . . . . . 6 𝑋 = dom 𝑂
87eqcomi 2746 . . . . 5 dom 𝑂 = 𝑋
98pweqi 4616 . . . 4 𝒫 dom 𝑂 = 𝒫 𝑋
109a1i 11 . . 3 (𝜑 → 𝒫 dom 𝑂 = 𝒫 𝑋)
116, 10eleqtrd 2843 . 2 (𝜑𝐴 ∈ 𝒫 𝑋)
12 elpwg 4603 . . 3 (𝐴𝑆 → (𝐴 ∈ 𝒫 𝑋𝐴𝑋))
131, 12syl 17 . 2 (𝜑 → (𝐴 ∈ 𝒫 𝑋𝐴𝑋))
1411, 13mpbid 232 1 (𝜑𝐴𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wral 3061  cdif 3948  cin 3950  wss 3951  𝒫 cpw 4600   cuni 4907  dom cdm 5685  cfv 6561  (class class class)co 7431   +𝑒 cxad 13152  OutMeascome 46504  CaraGenccaragen 46506
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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-caragen 46507
This theorem is referenced by:  caragenuncllem  46527  caragenuncl  46528
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