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Theorem caragenelss 46158
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 46152 . . . . 5 (𝜑 → (𝐴𝑆 ↔ (𝐴 ∈ 𝒫 dom 𝑂 ∧ ∀𝑥 ∈ 𝒫 dom 𝑂((𝑂‘(𝑥𝐴)) +𝑒 (𝑂‘(𝑥𝐴))) = (𝑂𝑥))))
51, 4mpbid 231 . . . 4 (𝜑 → (𝐴 ∈ 𝒫 dom 𝑂 ∧ ∀𝑥 ∈ 𝒫 dom 𝑂((𝑂‘(𝑥𝐴)) +𝑒 (𝑂‘(𝑥𝐴))) = (𝑂𝑥)))
65simpld 493 . . 3 (𝜑𝐴 ∈ 𝒫 dom 𝑂)
7 caragenelss.x . . . . . 6 𝑋 = dom 𝑂
87eqcomi 2735 . . . . 5 dom 𝑂 = 𝑋
98pweqi 4613 . . . 4 𝒫 dom 𝑂 = 𝒫 𝑋
109a1i 11 . . 3 (𝜑 → 𝒫 dom 𝑂 = 𝒫 𝑋)
116, 10eleqtrd 2828 . 2 (𝜑𝐴 ∈ 𝒫 𝑋)
12 elpwg 4600 . . 3 (𝐴𝑆 → (𝐴 ∈ 𝒫 𝑋𝐴𝑋))
131, 12syl 17 . 2 (𝜑 → (𝐴 ∈ 𝒫 𝑋𝐴𝑋))
1411, 13mpbid 231 1 (𝜑𝐴𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1534  wcel 2099  wral 3051  cdif 3943  cin 3945  wss 3946  𝒫 cpw 4597   cuni 4905  dom cdm 5674  cfv 6546  (class class class)co 7416   +𝑒 cxad 13138  OutMeascome 46146  CaraGenccaragen 46148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5296  ax-nul 5303  ax-pow 5361  ax-pr 5425  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-br 5146  df-opab 5208  df-mpt 5229  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-iota 6498  df-fun 6548  df-fv 6554  df-ov 7419  df-caragen 46149
This theorem is referenced by:  caragenuncllem  46169  caragenuncl  46170
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