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Theorem caragenel 46451
Description: Membership in the Caratheodory's construction. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypotheses
Ref Expression
caragenel.o (𝜑𝑂 ∈ OutMeas)
caragenel.s 𝑆 = (CaraGen‘𝑂)
Assertion
Ref Expression
caragenel (𝜑 → (𝐸𝑆 ↔ (𝐸 ∈ 𝒫 dom 𝑂 ∧ ∀𝑎 ∈ 𝒫 dom 𝑂((𝑂‘(𝑎𝐸)) +𝑒 (𝑂‘(𝑎𝐸))) = (𝑂𝑎))))
Distinct variable groups:   𝐸,𝑎   𝑂,𝑎
Allowed substitution hints:   𝜑(𝑎)   𝑆(𝑎)

Proof of Theorem caragenel
Dummy variable 𝑒 is distinct from all other variables.
StepHypRef Expression
1 caragenel.s . . . 4 𝑆 = (CaraGen‘𝑂)
2 caragenel.o . . . . 5 (𝜑𝑂 ∈ OutMeas)
3 caragenval 46449 . . . . 5 (𝑂 ∈ OutMeas → (CaraGen‘𝑂) = {𝑒 ∈ 𝒫 dom 𝑂 ∣ ∀𝑎 ∈ 𝒫 dom 𝑂((𝑂‘(𝑎𝑒)) +𝑒 (𝑂‘(𝑎𝑒))) = (𝑂𝑎)})
42, 3syl 17 . . . 4 (𝜑 → (CaraGen‘𝑂) = {𝑒 ∈ 𝒫 dom 𝑂 ∣ ∀𝑎 ∈ 𝒫 dom 𝑂((𝑂‘(𝑎𝑒)) +𝑒 (𝑂‘(𝑎𝑒))) = (𝑂𝑎)})
51, 4eqtrid 2787 . . 3 (𝜑𝑆 = {𝑒 ∈ 𝒫 dom 𝑂 ∣ ∀𝑎 ∈ 𝒫 dom 𝑂((𝑂‘(𝑎𝑒)) +𝑒 (𝑂‘(𝑎𝑒))) = (𝑂𝑎)})
65eleq2d 2825 . 2 (𝜑 → (𝐸𝑆𝐸 ∈ {𝑒 ∈ 𝒫 dom 𝑂 ∣ ∀𝑎 ∈ 𝒫 dom 𝑂((𝑂‘(𝑎𝑒)) +𝑒 (𝑂‘(𝑎𝑒))) = (𝑂𝑎)}))
7 ineq2 4222 . . . . . . . 8 (𝑒 = 𝐸 → (𝑎𝑒) = (𝑎𝐸))
87fveq2d 6911 . . . . . . 7 (𝑒 = 𝐸 → (𝑂‘(𝑎𝑒)) = (𝑂‘(𝑎𝐸)))
9 difeq2 4130 . . . . . . . 8 (𝑒 = 𝐸 → (𝑎𝑒) = (𝑎𝐸))
109fveq2d 6911 . . . . . . 7 (𝑒 = 𝐸 → (𝑂‘(𝑎𝑒)) = (𝑂‘(𝑎𝐸)))
118, 10oveq12d 7449 . . . . . 6 (𝑒 = 𝐸 → ((𝑂‘(𝑎𝑒)) +𝑒 (𝑂‘(𝑎𝑒))) = ((𝑂‘(𝑎𝐸)) +𝑒 (𝑂‘(𝑎𝐸))))
1211eqeq1d 2737 . . . . 5 (𝑒 = 𝐸 → (((𝑂‘(𝑎𝑒)) +𝑒 (𝑂‘(𝑎𝑒))) = (𝑂𝑎) ↔ ((𝑂‘(𝑎𝐸)) +𝑒 (𝑂‘(𝑎𝐸))) = (𝑂𝑎)))
1312ralbidv 3176 . . . 4 (𝑒 = 𝐸 → (∀𝑎 ∈ 𝒫 dom 𝑂((𝑂‘(𝑎𝑒)) +𝑒 (𝑂‘(𝑎𝑒))) = (𝑂𝑎) ↔ ∀𝑎 ∈ 𝒫 dom 𝑂((𝑂‘(𝑎𝐸)) +𝑒 (𝑂‘(𝑎𝐸))) = (𝑂𝑎)))
1413elrab 3695 . . 3 (𝐸 ∈ {𝑒 ∈ 𝒫 dom 𝑂 ∣ ∀𝑎 ∈ 𝒫 dom 𝑂((𝑂‘(𝑎𝑒)) +𝑒 (𝑂‘(𝑎𝑒))) = (𝑂𝑎)} ↔ (𝐸 ∈ 𝒫 dom 𝑂 ∧ ∀𝑎 ∈ 𝒫 dom 𝑂((𝑂‘(𝑎𝐸)) +𝑒 (𝑂‘(𝑎𝐸))) = (𝑂𝑎)))
1514a1i 11 . 2 (𝜑 → (𝐸 ∈ {𝑒 ∈ 𝒫 dom 𝑂 ∣ ∀𝑎 ∈ 𝒫 dom 𝑂((𝑂‘(𝑎𝑒)) +𝑒 (𝑂‘(𝑎𝑒))) = (𝑂𝑎)} ↔ (𝐸 ∈ 𝒫 dom 𝑂 ∧ ∀𝑎 ∈ 𝒫 dom 𝑂((𝑂‘(𝑎𝐸)) +𝑒 (𝑂‘(𝑎𝐸))) = (𝑂𝑎))))
166, 15bitrd 279 1 (𝜑 → (𝐸𝑆 ↔ (𝐸 ∈ 𝒫 dom 𝑂 ∧ ∀𝑎 ∈ 𝒫 dom 𝑂((𝑂‘(𝑎𝐸)) +𝑒 (𝑂‘(𝑎𝐸))) = (𝑂𝑎))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wral 3059  {crab 3433  cdif 3960  cin 3962  𝒫 cpw 4605   cuni 4912  dom cdm 5689  cfv 6563  (class class class)co 7431   +𝑒 cxad 13150  OutMeascome 46445  CaraGenccaragen 46447
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-caragen 46448
This theorem is referenced by:  caragensplit  46456  caragenelss  46457  carageneld  46458  caragendifcl  46470  isvonmbl  46594
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