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Theorem caragenelss 39391
 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 39385 . . . . 5 (𝜑 → (𝐴𝑆 ↔ (𝐴 ∈ 𝒫 dom 𝑂 ∧ ∀𝑥 ∈ 𝒫 dom 𝑂((𝑂‘(𝑥𝐴)) +𝑒 (𝑂‘(𝑥𝐴))) = (𝑂𝑥))))
51, 4mpbid 221 . . . 4 (𝜑 → (𝐴 ∈ 𝒫 dom 𝑂 ∧ ∀𝑥 ∈ 𝒫 dom 𝑂((𝑂‘(𝑥𝐴)) +𝑒 (𝑂‘(𝑥𝐴))) = (𝑂𝑥)))
65simpld 474 . . 3 (𝜑𝐴 ∈ 𝒫 dom 𝑂)
7 caragenelss.x . . . . . 6 𝑋 = dom 𝑂
87eqcomi 2619 . . . . 5 dom 𝑂 = 𝑋
98pweqi 4112 . . . 4 𝒫 dom 𝑂 = 𝒫 𝑋
109a1i 11 . . 3 (𝜑 → 𝒫 dom 𝑂 = 𝒫 𝑋)
116, 10eleqtrd 2690 . 2 (𝜑𝐴 ∈ 𝒫 𝑋)
12 elpwg 4116 . . 3 (𝐴𝑆 → (𝐴 ∈ 𝒫 𝑋𝐴𝑋))
131, 12syl 17 . 2 (𝜑 → (𝐴 ∈ 𝒫 𝑋𝐴𝑋))
1411, 13mpbid 221 1 (𝜑𝐴𝑋)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ∖ cdif 3537   ∩ cin 3539   ⊆ wss 3540  𝒫 cpw 4108  ∪ cuni 4372  dom cdm 5038  ‘cfv 5804  (class class class)co 6549   +𝑒 cxad 11820  OutMeascome 39379  CaraGenccaragen 39381 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-caragen 39382 This theorem is referenced by:  caragenuncllem  39402  caragenuncl  39403
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