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Theorem caragenel 41501
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 41499 . . . . 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, 4syl5eq 2873 . . 3 (𝜑𝑆 = {𝑒 ∈ 𝒫 dom 𝑂 ∣ ∀𝑎 ∈ 𝒫 dom 𝑂((𝑂‘(𝑎𝑒)) +𝑒 (𝑂‘(𝑎𝑒))) = (𝑂𝑎)})
65eleq2d 2892 . 2 (𝜑 → (𝐸𝑆𝐸 ∈ {𝑒 ∈ 𝒫 dom 𝑂 ∣ ∀𝑎 ∈ 𝒫 dom 𝑂((𝑂‘(𝑎𝑒)) +𝑒 (𝑂‘(𝑎𝑒))) = (𝑂𝑎)}))
7 ineq2 4037 . . . . . . . 8 (𝑒 = 𝐸 → (𝑎𝑒) = (𝑎𝐸))
87fveq2d 6441 . . . . . . 7 (𝑒 = 𝐸 → (𝑂‘(𝑎𝑒)) = (𝑂‘(𝑎𝐸)))
9 difeq2 3951 . . . . . . . 8 (𝑒 = 𝐸 → (𝑎𝑒) = (𝑎𝐸))
109fveq2d 6441 . . . . . . 7 (𝑒 = 𝐸 → (𝑂‘(𝑎𝑒)) = (𝑂‘(𝑎𝐸)))
118, 10oveq12d 6928 . . . . . 6 (𝑒 = 𝐸 → ((𝑂‘(𝑎𝑒)) +𝑒 (𝑂‘(𝑎𝑒))) = ((𝑂‘(𝑎𝐸)) +𝑒 (𝑂‘(𝑎𝐸))))
1211eqeq1d 2827 . . . . 5 (𝑒 = 𝐸 → (((𝑂‘(𝑎𝑒)) +𝑒 (𝑂‘(𝑎𝑒))) = (𝑂𝑎) ↔ ((𝑂‘(𝑎𝐸)) +𝑒 (𝑂‘(𝑎𝐸))) = (𝑂𝑎)))
1312ralbidv 3195 . . . 4 (𝑒 = 𝐸 → (∀𝑎 ∈ 𝒫 dom 𝑂((𝑂‘(𝑎𝑒)) +𝑒 (𝑂‘(𝑎𝑒))) = (𝑂𝑎) ↔ ∀𝑎 ∈ 𝒫 dom 𝑂((𝑂‘(𝑎𝐸)) +𝑒 (𝑂‘(𝑎𝐸))) = (𝑂𝑎)))
1413elrab 3585 . . 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 271 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 198  wa 386   = wceq 1656  wcel 2164  wral 3117  {crab 3121  cdif 3795  cin 3797  𝒫 cpw 4380   cuni 4660  dom cdm 5346  cfv 6127  (class class class)co 6910   +𝑒 cxad 12237  OutMeascome 41495  CaraGenccaragen 41497
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 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214
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 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-iota 6090  df-fun 6129  df-fv 6135  df-ov 6913  df-caragen 41498
This theorem is referenced by:  caragensplit  41506  caragenelss  41507  carageneld  41508  caragendifcl  41520  isvonmbl  41644
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