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Theorem carageneld 40020
 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 4134 . . . 4 𝒫 𝑋 = 𝒫 dom 𝑂
41, 3syl6eleq 2708 . . 3 (𝜑𝐸 ∈ 𝒫 dom 𝑂)
5 simpl 473 . . . . 5 ((𝜑𝑎 ∈ 𝒫 dom 𝑂) → 𝜑)
63eleq2i 2690 . . . . . . . 8 (𝑎 ∈ 𝒫 𝑋𝑎 ∈ 𝒫 dom 𝑂)
76bicomi 214 . . . . . . 7 (𝑎 ∈ 𝒫 dom 𝑂𝑎 ∈ 𝒫 𝑋)
87biimpi 206 . . . . . 6 (𝑎 ∈ 𝒫 dom 𝑂𝑎 ∈ 𝒫 𝑋)
98adantl 482 . . . . 5 ((𝜑𝑎 ∈ 𝒫 dom 𝑂) → 𝑎 ∈ 𝒫 𝑋)
10 carageneld.a . . . . 5 ((𝜑𝑎 ∈ 𝒫 𝑋) → ((𝑂‘(𝑎𝐸)) +𝑒 (𝑂‘(𝑎𝐸))) = (𝑂𝑎))
115, 9, 10syl2anc 692 . . . 4 ((𝜑𝑎 ∈ 𝒫 dom 𝑂) → ((𝑂‘(𝑎𝐸)) +𝑒 (𝑂‘(𝑎𝐸))) = (𝑂𝑎))
1211ralrimiva 2960 . . 3 (𝜑 → ∀𝑎 ∈ 𝒫 dom 𝑂((𝑂‘(𝑎𝐸)) +𝑒 (𝑂‘(𝑎𝐸))) = (𝑂𝑎))
134, 12jca 554 . 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 40013 . 2 (𝜑 → (𝐸𝑆 ↔ (𝐸 ∈ 𝒫 dom 𝑂 ∧ ∀𝑎 ∈ 𝒫 dom 𝑂((𝑂‘(𝑎𝐸)) +𝑒 (𝑂‘(𝑎𝐸))) = (𝑂𝑎))))
1713, 16mpbird 247 1 (𝜑𝐸𝑆)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   = wceq 1480   ∈ wcel 1987  ∀wral 2907   ∖ cdif 3552   ∩ cin 3554  𝒫 cpw 4130  ∪ cuni 4402  dom cdm 5074  ‘cfv 5847  (class class class)co 6604   +𝑒 cxad 11888  OutMeascome 40007  CaraGenccaragen 40009 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-iota 5810  df-fun 5849  df-fv 5855  df-ov 6607  df-caragen 40010 This theorem is referenced by:  caragen0  40024  caragenunidm  40026  caragenuncl  40031  caragendifcl  40032  carageniuncl  40041  caragenel2d  40050
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