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Theorem caragenelss 45203
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 (πœ‘ β†’ 𝑂 ∈ OutMeas)
caragenelss.s 𝑆 = (CaraGenβ€˜π‘‚)
caragenelss.a (πœ‘ β†’ 𝐴 ∈ 𝑆)
caragenelss.x 𝑋 = βˆͺ dom 𝑂
Assertion
Ref Expression
caragenelss (πœ‘ β†’ 𝐴 βŠ† 𝑋)

Proof of Theorem caragenelss
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 caragenelss.a . . . . 5 (πœ‘ β†’ 𝐴 ∈ 𝑆)
2 caragenelss.o . . . . . 6 (πœ‘ β†’ 𝑂 ∈ OutMeas)
3 caragenelss.s . . . . . 6 𝑆 = (CaraGenβ€˜π‘‚)
42, 3caragenel 45197 . . . . 5 (πœ‘ β†’ (𝐴 ∈ 𝑆 ↔ (𝐴 ∈ 𝒫 βˆͺ dom 𝑂 ∧ βˆ€π‘₯ ∈ 𝒫 βˆͺ dom 𝑂((π‘‚β€˜(π‘₯ ∩ 𝐴)) +𝑒 (π‘‚β€˜(π‘₯ βˆ– 𝐴))) = (π‘‚β€˜π‘₯))))
51, 4mpbid 231 . . . 4 (πœ‘ β†’ (𝐴 ∈ 𝒫 βˆͺ dom 𝑂 ∧ βˆ€π‘₯ ∈ 𝒫 βˆͺ dom 𝑂((π‘‚β€˜(π‘₯ ∩ 𝐴)) +𝑒 (π‘‚β€˜(π‘₯ βˆ– 𝐴))) = (π‘‚β€˜π‘₯)))
65simpld 495 . . 3 (πœ‘ β†’ 𝐴 ∈ 𝒫 βˆͺ dom 𝑂)
7 caragenelss.x . . . . . 6 𝑋 = βˆͺ dom 𝑂
87eqcomi 2741 . . . . 5 βˆͺ dom 𝑂 = 𝑋
98pweqi 4617 . . . 4 𝒫 βˆͺ dom 𝑂 = 𝒫 𝑋
109a1i 11 . . 3 (πœ‘ β†’ 𝒫 βˆͺ dom 𝑂 = 𝒫 𝑋)
116, 10eleqtrd 2835 . 2 (πœ‘ β†’ 𝐴 ∈ 𝒫 𝑋)
12 elpwg 4604 . . 3 (𝐴 ∈ 𝑆 β†’ (𝐴 ∈ 𝒫 𝑋 ↔ 𝐴 βŠ† 𝑋))
131, 12syl 17 . 2 (πœ‘ β†’ (𝐴 ∈ 𝒫 𝑋 ↔ 𝐴 βŠ† 𝑋))
1411, 13mpbid 231 1 (πœ‘ β†’ 𝐴 βŠ† 𝑋)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061   βˆ– cdif 3944   ∩ cin 3946   βŠ† wss 3947  π’« cpw 4601  βˆͺ cuni 4907  dom cdm 5675  β€˜cfv 6540  (class class class)co 7405   +𝑒 cxad 13086  OutMeascome 45191  CaraGenccaragen 45193
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-iota 6492  df-fun 6542  df-fv 6548  df-ov 7408  df-caragen 45194
This theorem is referenced by:  caragenuncllem  45214  caragenuncl  45215
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