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Theorem caragenel 40042
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 40040 . . . . 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 2667 . . 3 (𝜑𝑆 = {𝑒 ∈ 𝒫 dom 𝑂 ∣ ∀𝑎 ∈ 𝒫 dom 𝑂((𝑂‘(𝑎𝑒)) +𝑒 (𝑂‘(𝑎𝑒))) = (𝑂𝑎)})
65eleq2d 2684 . 2 (𝜑 → (𝐸𝑆𝐸 ∈ {𝑒 ∈ 𝒫 dom 𝑂 ∣ ∀𝑎 ∈ 𝒫 dom 𝑂((𝑂‘(𝑎𝑒)) +𝑒 (𝑂‘(𝑎𝑒))) = (𝑂𝑎)}))
7 ineq2 3791 . . . . . . . 8 (𝑒 = 𝐸 → (𝑎𝑒) = (𝑎𝐸))
87fveq2d 6157 . . . . . . 7 (𝑒 = 𝐸 → (𝑂‘(𝑎𝑒)) = (𝑂‘(𝑎𝐸)))
9 difeq2 3705 . . . . . . . 8 (𝑒 = 𝐸 → (𝑎𝑒) = (𝑎𝐸))
109fveq2d 6157 . . . . . . 7 (𝑒 = 𝐸 → (𝑂‘(𝑎𝑒)) = (𝑂‘(𝑎𝐸)))
118, 10oveq12d 6628 . . . . . 6 (𝑒 = 𝐸 → ((𝑂‘(𝑎𝑒)) +𝑒 (𝑂‘(𝑎𝑒))) = ((𝑂‘(𝑎𝐸)) +𝑒 (𝑂‘(𝑎𝐸))))
1211eqeq1d 2623 . . . . 5 (𝑒 = 𝐸 → (((𝑂‘(𝑎𝑒)) +𝑒 (𝑂‘(𝑎𝑒))) = (𝑂𝑎) ↔ ((𝑂‘(𝑎𝐸)) +𝑒 (𝑂‘(𝑎𝐸))) = (𝑂𝑎)))
1312ralbidv 2981 . . . 4 (𝑒 = 𝐸 → (∀𝑎 ∈ 𝒫 dom 𝑂((𝑂‘(𝑎𝑒)) +𝑒 (𝑂‘(𝑎𝑒))) = (𝑂𝑎) ↔ ∀𝑎 ∈ 𝒫 dom 𝑂((𝑂‘(𝑎𝐸)) +𝑒 (𝑂‘(𝑎𝐸))) = (𝑂𝑎)))
1413elrab 3350 . . 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 268 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 196  wa 384   = wceq 1480  wcel 1987  wral 2907  {crab 2911  cdif 3556  cin 3558  𝒫 cpw 4135   cuni 4407  dom cdm 5079  cfv 5852  (class class class)co 6610   +𝑒 cxad 11896  OutMeascome 40036  CaraGenccaragen 40038
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-iota 5815  df-fun 5854  df-fv 5860  df-ov 6613  df-caragen 40039
This theorem is referenced by:  caragensplit  40047  caragenelss  40048  carageneld  40049  caragendifcl  40061  isvonmbl  40185
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