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Theorem caragenelss 41239
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 41233 . . . . 5 (𝜑 → (𝐴𝑆 ↔ (𝐴 ∈ 𝒫 dom 𝑂 ∧ ∀𝑥 ∈ 𝒫 dom 𝑂((𝑂‘(𝑥𝐴)) +𝑒 (𝑂‘(𝑥𝐴))) = (𝑂𝑥))))
51, 4mpbid 222 . . . 4 (𝜑 → (𝐴 ∈ 𝒫 dom 𝑂 ∧ ∀𝑥 ∈ 𝒫 dom 𝑂((𝑂‘(𝑥𝐴)) +𝑒 (𝑂‘(𝑥𝐴))) = (𝑂𝑥)))
65simpld 477 . . 3 (𝜑𝐴 ∈ 𝒫 dom 𝑂)
7 caragenelss.x . . . . . 6 𝑋 = dom 𝑂
87eqcomi 2769 . . . . 5 dom 𝑂 = 𝑋
98pweqi 4306 . . . 4 𝒫 dom 𝑂 = 𝒫 𝑋
109a1i 11 . . 3 (𝜑 → 𝒫 dom 𝑂 = 𝒫 𝑋)
116, 10eleqtrd 2841 . 2 (𝜑𝐴 ∈ 𝒫 𝑋)
12 elpwg 4310 . . 3 (𝐴𝑆 → (𝐴 ∈ 𝒫 𝑋𝐴𝑋))
131, 12syl 17 . 2 (𝜑 → (𝐴 ∈ 𝒫 𝑋𝐴𝑋))
1411, 13mpbid 222 1 (𝜑𝐴𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  wral 3050  cdif 3712  cin 3714  wss 3715  𝒫 cpw 4302   cuni 4588  dom cdm 5266  cfv 6049  (class class class)co 6814   +𝑒 cxad 12157  OutMeascome 41227  CaraGenccaragen 41229
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-iota 6012  df-fun 6051  df-fv 6057  df-ov 6817  df-caragen 41230
This theorem is referenced by:  caragenuncllem  41250  caragenuncl  41251
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