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Theorem caragenelss 43006
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 43000 . . . . 5 (𝜑 → (𝐴𝑆 ↔ (𝐴 ∈ 𝒫 dom 𝑂 ∧ ∀𝑥 ∈ 𝒫 dom 𝑂((𝑂‘(𝑥𝐴)) +𝑒 (𝑂‘(𝑥𝐴))) = (𝑂𝑥))))
51, 4mpbid 235 . . . 4 (𝜑 → (𝐴 ∈ 𝒫 dom 𝑂 ∧ ∀𝑥 ∈ 𝒫 dom 𝑂((𝑂‘(𝑥𝐴)) +𝑒 (𝑂‘(𝑥𝐴))) = (𝑂𝑥)))
65simpld 498 . . 3 (𝜑𝐴 ∈ 𝒫 dom 𝑂)
7 caragenelss.x . . . . . 6 𝑋 = dom 𝑂
87eqcomi 2833 . . . . 5 dom 𝑂 = 𝑋
98pweqi 4540 . . . 4 𝒫 dom 𝑂 = 𝒫 𝑋
109a1i 11 . . 3 (𝜑 → 𝒫 dom 𝑂 = 𝒫 𝑋)
116, 10eleqtrd 2918 . 2 (𝜑𝐴 ∈ 𝒫 𝑋)
12 elpwg 4525 . . 3 (𝐴𝑆 → (𝐴 ∈ 𝒫 𝑋𝐴𝑋))
131, 12syl 17 . 2 (𝜑 → (𝐴 ∈ 𝒫 𝑋𝐴𝑋))
1411, 13mpbid 235 1 (𝜑𝐴𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  wral 3133  cdif 3916  cin 3918  wss 3919  𝒫 cpw 4522   cuni 4825  dom cdm 5543  cfv 6344  (class class class)co 7146   +𝑒 cxad 12500  OutMeascome 42994  CaraGenccaragen 42996
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7452
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4826  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-iota 6303  df-fun 6346  df-fv 6352  df-ov 7149  df-caragen 42997
This theorem is referenced by:  caragenuncllem  43017  caragenuncl  43018
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