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Theorem carageneld 46517
Description: Membership in the Caratheodory's construction. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypotheses
Ref Expression
carageneld.o (𝜑𝑂 ∈ OutMeas)
carageneld.x 𝑋 = dom 𝑂
carageneld.s 𝑆 = (CaraGen‘𝑂)
carageneld.e (𝜑𝐸 ∈ 𝒫 𝑋)
carageneld.a ((𝜑𝑎 ∈ 𝒫 𝑋) → ((𝑂‘(𝑎𝐸)) +𝑒 (𝑂‘(𝑎𝐸))) = (𝑂𝑎))
Assertion
Ref Expression
carageneld (𝜑𝐸𝑆)
Distinct variable groups:   𝐸,𝑎   𝑂,𝑎   𝜑,𝑎
Allowed substitution hints:   𝑆(𝑎)   𝑋(𝑎)

Proof of Theorem carageneld
StepHypRef Expression
1 carageneld.e . . . 4 (𝜑𝐸 ∈ 𝒫 𝑋)
2 carageneld.x . . . . 5 𝑋 = dom 𝑂
32pweqi 4616 . . . 4 𝒫 𝑋 = 𝒫 dom 𝑂
41, 3eleqtrdi 2851 . . 3 (𝜑𝐸 ∈ 𝒫 dom 𝑂)
5 simpl 482 . . . . 5 ((𝜑𝑎 ∈ 𝒫 dom 𝑂) → 𝜑)
63eleq2i 2833 . . . . . . . 8 (𝑎 ∈ 𝒫 𝑋𝑎 ∈ 𝒫 dom 𝑂)
76bicomi 224 . . . . . . 7 (𝑎 ∈ 𝒫 dom 𝑂𝑎 ∈ 𝒫 𝑋)
87biimpi 216 . . . . . 6 (𝑎 ∈ 𝒫 dom 𝑂𝑎 ∈ 𝒫 𝑋)
98adantl 481 . . . . 5 ((𝜑𝑎 ∈ 𝒫 dom 𝑂) → 𝑎 ∈ 𝒫 𝑋)
10 carageneld.a . . . . 5 ((𝜑𝑎 ∈ 𝒫 𝑋) → ((𝑂‘(𝑎𝐸)) +𝑒 (𝑂‘(𝑎𝐸))) = (𝑂𝑎))
115, 9, 10syl2anc 584 . . . 4 ((𝜑𝑎 ∈ 𝒫 dom 𝑂) → ((𝑂‘(𝑎𝐸)) +𝑒 (𝑂‘(𝑎𝐸))) = (𝑂𝑎))
1211ralrimiva 3146 . . 3 (𝜑 → ∀𝑎 ∈ 𝒫 dom 𝑂((𝑂‘(𝑎𝐸)) +𝑒 (𝑂‘(𝑎𝐸))) = (𝑂𝑎))
134, 12jca 511 . 2 (𝜑 → (𝐸 ∈ 𝒫 dom 𝑂 ∧ ∀𝑎 ∈ 𝒫 dom 𝑂((𝑂‘(𝑎𝐸)) +𝑒 (𝑂‘(𝑎𝐸))) = (𝑂𝑎)))
14 carageneld.o . . 3 (𝜑𝑂 ∈ OutMeas)
15 carageneld.s . . 3 𝑆 = (CaraGen‘𝑂)
1614, 15caragenel 46510 . 2 (𝜑 → (𝐸𝑆 ↔ (𝐸 ∈ 𝒫 dom 𝑂 ∧ ∀𝑎 ∈ 𝒫 dom 𝑂((𝑂‘(𝑎𝐸)) +𝑒 (𝑂‘(𝑎𝐸))) = (𝑂𝑎))))
1713, 16mpbird 257 1 (𝜑𝐸𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  wral 3061  cdif 3948  cin 3950  𝒫 cpw 4600   cuni 4907  dom cdm 5685  cfv 6561  (class class class)co 7431   +𝑒 cxad 13152  OutMeascome 46504  CaraGenccaragen 46506
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-caragen 46507
This theorem is referenced by:  caragen0  46521  caragenunidm  46523  caragenuncl  46528  caragendifcl  46529  carageniuncl  46538  caragenel2d  46547
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