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Theorem carageneld 46458
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 4621 . . . 4 𝒫 𝑋 = 𝒫 dom 𝑂
41, 3eleqtrdi 2849 . . 3 (𝜑𝐸 ∈ 𝒫 dom 𝑂)
5 simpl 482 . . . . 5 ((𝜑𝑎 ∈ 𝒫 dom 𝑂) → 𝜑)
63eleq2i 2831 . . . . . . . 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 3144 . . 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 46451 . 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 1537  wcel 2106  wral 3059  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:  caragen0  46462  caragenunidm  46464  caragenuncl  46469  caragendifcl  46470  carageniuncl  46479  caragenel2d  46488
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