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Theorem carageneld 41503
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 4382 . . . 4 𝒫 𝑋 = 𝒫 dom 𝑂
41, 3syl6eleq 2916 . . 3 (𝜑𝐸 ∈ 𝒫 dom 𝑂)
5 simpl 476 . . . . 5 ((𝜑𝑎 ∈ 𝒫 dom 𝑂) → 𝜑)
63eleq2i 2898 . . . . . . . 8 (𝑎 ∈ 𝒫 𝑋𝑎 ∈ 𝒫 dom 𝑂)
76bicomi 216 . . . . . . 7 (𝑎 ∈ 𝒫 dom 𝑂𝑎 ∈ 𝒫 𝑋)
87biimpi 208 . . . . . 6 (𝑎 ∈ 𝒫 dom 𝑂𝑎 ∈ 𝒫 𝑋)
98adantl 475 . . . . 5 ((𝜑𝑎 ∈ 𝒫 dom 𝑂) → 𝑎 ∈ 𝒫 𝑋)
10 carageneld.a . . . . 5 ((𝜑𝑎 ∈ 𝒫 𝑋) → ((𝑂‘(𝑎𝐸)) +𝑒 (𝑂‘(𝑎𝐸))) = (𝑂𝑎))
115, 9, 10syl2anc 579 . . . 4 ((𝜑𝑎 ∈ 𝒫 dom 𝑂) → ((𝑂‘(𝑎𝐸)) +𝑒 (𝑂‘(𝑎𝐸))) = (𝑂𝑎))
1211ralrimiva 3175 . . 3 (𝜑 → ∀𝑎 ∈ 𝒫 dom 𝑂((𝑂‘(𝑎𝐸)) +𝑒 (𝑂‘(𝑎𝐸))) = (𝑂𝑎))
134, 12jca 507 . 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 41496 . 2 (𝜑 → (𝐸𝑆 ↔ (𝐸 ∈ 𝒫 dom 𝑂 ∧ ∀𝑎 ∈ 𝒫 dom 𝑂((𝑂‘(𝑎𝐸)) +𝑒 (𝑂‘(𝑎𝐸))) = (𝑂𝑎))))
1713, 16mpbird 249 1 (𝜑𝐸𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1656  wcel 2164  wral 3117  cdif 3795  cin 3797  𝒫 cpw 4378   cuni 4658  dom cdm 5342  cfv 6123  (class class class)co 6905   +𝑒 cxad 12230  OutMeascome 41490  CaraGenccaragen 41492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-iota 6086  df-fun 6125  df-fv 6131  df-ov 6908  df-caragen 41493
This theorem is referenced by:  caragen0  41507  caragenunidm  41509  caragenuncl  41514  caragendifcl  41515  carageniuncl  41524  caragenel2d  41533
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