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Theorem carageneld 41222
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 4306 . . . 4 𝒫 𝑋 = 𝒫 dom 𝑂
41, 3syl6eleq 2849 . . 3 (𝜑𝐸 ∈ 𝒫 dom 𝑂)
5 simpl 474 . . . . 5 ((𝜑𝑎 ∈ 𝒫 dom 𝑂) → 𝜑)
63eleq2i 2831 . . . . . . . 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 473 . . . . 5 ((𝜑𝑎 ∈ 𝒫 dom 𝑂) → 𝑎 ∈ 𝒫 𝑋)
10 carageneld.a . . . . 5 ((𝜑𝑎 ∈ 𝒫 𝑋) → ((𝑂‘(𝑎𝐸)) +𝑒 (𝑂‘(𝑎𝐸))) = (𝑂𝑎))
115, 9, 10syl2anc 696 . . . 4 ((𝜑𝑎 ∈ 𝒫 dom 𝑂) → ((𝑂‘(𝑎𝐸)) +𝑒 (𝑂‘(𝑎𝐸))) = (𝑂𝑎))
1211ralrimiva 3104 . . 3 (𝜑 → ∀𝑎 ∈ 𝒫 dom 𝑂((𝑂‘(𝑎𝐸)) +𝑒 (𝑂‘(𝑎𝐸))) = (𝑂𝑎))
134, 12jca 555 . 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 41215 . 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 383   = wceq 1632  wcel 2139  wral 3050  cdif 3712  cin 3714  𝒫 cpw 4302   cuni 4588  dom cdm 5266  cfv 6049  (class class class)co 6813   +𝑒 cxad 12137  OutMeascome 41209  CaraGenccaragen 41211
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-iota 6012  df-fun 6051  df-fv 6057  df-ov 6816  df-caragen 41212
This theorem is referenced by:  caragen0  41226  caragenunidm  41228  caragenuncl  41233  caragendifcl  41234  carageniuncl  41243  caragenel2d  41252
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