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Theorem caragenel 42645
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 42643 . . . . 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 2872 . . 3 (𝜑𝑆 = {𝑒 ∈ 𝒫 dom 𝑂 ∣ ∀𝑎 ∈ 𝒫 dom 𝑂((𝑂‘(𝑎𝑒)) +𝑒 (𝑂‘(𝑎𝑒))) = (𝑂𝑎)})
65eleq2d 2902 . 2 (𝜑 → (𝐸𝑆𝐸 ∈ {𝑒 ∈ 𝒫 dom 𝑂 ∣ ∀𝑎 ∈ 𝒫 dom 𝑂((𝑂‘(𝑎𝑒)) +𝑒 (𝑂‘(𝑎𝑒))) = (𝑂𝑎)}))
7 ineq2 4186 . . . . . . . 8 (𝑒 = 𝐸 → (𝑎𝑒) = (𝑎𝐸))
87fveq2d 6670 . . . . . . 7 (𝑒 = 𝐸 → (𝑂‘(𝑎𝑒)) = (𝑂‘(𝑎𝐸)))
9 difeq2 4096 . . . . . . . 8 (𝑒 = 𝐸 → (𝑎𝑒) = (𝑎𝐸))
109fveq2d 6670 . . . . . . 7 (𝑒 = 𝐸 → (𝑂‘(𝑎𝑒)) = (𝑂‘(𝑎𝐸)))
118, 10oveq12d 7169 . . . . . 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 3201 . . . 4 (𝑒 = 𝐸 → (∀𝑎 ∈ 𝒫 dom 𝑂((𝑂‘(𝑎𝑒)) +𝑒 (𝑂‘(𝑎𝑒))) = (𝑂𝑎) ↔ ∀𝑎 ∈ 𝒫 dom 𝑂((𝑂‘(𝑎𝐸)) +𝑒 (𝑂‘(𝑎𝐸))) = (𝑂𝑎)))
1413elrab 3683 . . 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 280 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 207  wa 396   = wceq 1530  wcel 2107  wral 3142  {crab 3146  cdif 3936  cin 3938  𝒫 cpw 4541   cuni 4836  dom cdm 5553  cfv 6351  (class class class)co 7151   +𝑒 cxad 12498  OutMeascome 42639  CaraGenccaragen 42641
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-iota 6311  df-fun 6353  df-fv 6359  df-ov 7154  df-caragen 42642
This theorem is referenced by:  caragensplit  42650  caragenelss  42651  carageneld  42652  caragendifcl  42664  isvonmbl  42788
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