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Theorem caragenelss 44816
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 44810 . . . . 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 496 . . 3 (πœ‘ β†’ 𝐴 ∈ 𝒫 βˆͺ dom 𝑂)
7 caragenelss.x . . . . . 6 𝑋 = βˆͺ dom 𝑂
87eqcomi 2746 . . . . 5 βˆͺ dom 𝑂 = 𝑋
98pweqi 4581 . . . 4 𝒫 βˆͺ dom 𝑂 = 𝒫 𝑋
109a1i 11 . . 3 (πœ‘ β†’ 𝒫 βˆͺ dom 𝑂 = 𝒫 𝑋)
116, 10eleqtrd 2840 . 2 (πœ‘ β†’ 𝐴 ∈ 𝒫 𝑋)
12 elpwg 4568 . . 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 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3065   βˆ– cdif 3912   ∩ cin 3914   βŠ† wss 3915  π’« cpw 4565  βˆͺ cuni 4870  dom cdm 5638  β€˜cfv 6501  (class class class)co 7362   +𝑒 cxad 13038  OutMeascome 44804  CaraGenccaragen 44806
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-iota 6453  df-fun 6503  df-fv 6509  df-ov 7365  df-caragen 44807
This theorem is referenced by:  caragenuncllem  44827  caragenuncl  44828
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