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Theorem elunirnmbfm 34253
Description: The property of being a measurable function. (Contributed by Thierry Arnoux, 23-Jan-2017.)
Assertion
Ref Expression
elunirnmbfm (𝐹 ran MblFnM ↔ ∃𝑠 ran sigAlgebra∃𝑡 ran sigAlgebra(𝐹 ∈ ( 𝑡m 𝑠) ∧ ∀𝑥𝑡 (𝐹𝑥) ∈ 𝑠))
Distinct variable group:   𝑡,𝑠,𝐹,𝑥

Proof of Theorem elunirnmbfm
Dummy variables 𝑓 𝑎 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-mbfm 34251 . . . . 5 MblFnM = (𝑠 ran sigAlgebra, 𝑡 ran sigAlgebra ↦ {𝑓 ∈ ( 𝑡m 𝑠) ∣ ∀𝑥𝑡 (𝑓𝑥) ∈ 𝑠})
21mpofun 7557 . . . 4 Fun MblFnM
3 elunirn 7271 . . . 4 (Fun MblFnM → (𝐹 ran MblFnM ↔ ∃𝑎 ∈ dom MblFnM𝐹 ∈ (MblFnM‘𝑎)))
42, 3ax-mp 5 . . 3 (𝐹 ran MblFnM ↔ ∃𝑎 ∈ dom MblFnM𝐹 ∈ (MblFnM‘𝑎))
5 ovex 7464 . . . . . 6 ( 𝑡m 𝑠) ∈ V
65rabex 5339 . . . . 5 {𝑓 ∈ ( 𝑡m 𝑠) ∣ ∀𝑥𝑡 (𝑓𝑥) ∈ 𝑠} ∈ V
71, 6dmmpo 8096 . . . 4 dom MblFnM = ( ran sigAlgebra × ran sigAlgebra)
87rexeqi 3325 . . 3 (∃𝑎 ∈ dom MblFnM𝐹 ∈ (MblFnM‘𝑎) ↔ ∃𝑎 ∈ ( ran sigAlgebra × ran sigAlgebra)𝐹 ∈ (MblFnM‘𝑎))
9 fveq2 6906 . . . . . 6 (𝑎 = ⟨𝑠, 𝑡⟩ → (MblFnM‘𝑎) = (MblFnM‘⟨𝑠, 𝑡⟩))
10 df-ov 7434 . . . . . 6 (𝑠MblFnM𝑡) = (MblFnM‘⟨𝑠, 𝑡⟩)
119, 10eqtr4di 2795 . . . . 5 (𝑎 = ⟨𝑠, 𝑡⟩ → (MblFnM‘𝑎) = (𝑠MblFnM𝑡))
1211eleq2d 2827 . . . 4 (𝑎 = ⟨𝑠, 𝑡⟩ → (𝐹 ∈ (MblFnM‘𝑎) ↔ 𝐹 ∈ (𝑠MblFnM𝑡)))
1312rexxp 5853 . . 3 (∃𝑎 ∈ ( ran sigAlgebra × ran sigAlgebra)𝐹 ∈ (MblFnM‘𝑎) ↔ ∃𝑠 ran sigAlgebra∃𝑡 ran sigAlgebra𝐹 ∈ (𝑠MblFnM𝑡))
144, 8, 133bitri 297 . 2 (𝐹 ran MblFnM ↔ ∃𝑠 ran sigAlgebra∃𝑡 ran sigAlgebra𝐹 ∈ (𝑠MblFnM𝑡))
15 simpl 482 . . . 4 ((𝑠 ran sigAlgebra ∧ 𝑡 ran sigAlgebra) → 𝑠 ran sigAlgebra)
16 simpr 484 . . . 4 ((𝑠 ran sigAlgebra ∧ 𝑡 ran sigAlgebra) → 𝑡 ran sigAlgebra)
1715, 16ismbfm 34252 . . 3 ((𝑠 ran sigAlgebra ∧ 𝑡 ran sigAlgebra) → (𝐹 ∈ (𝑠MblFnM𝑡) ↔ (𝐹 ∈ ( 𝑡m 𝑠) ∧ ∀𝑥𝑡 (𝐹𝑥) ∈ 𝑠)))
18172rexbiia 3218 . 2 (∃𝑠 ran sigAlgebra∃𝑡 ran sigAlgebra𝐹 ∈ (𝑠MblFnM𝑡) ↔ ∃𝑠 ran sigAlgebra∃𝑡 ran sigAlgebra(𝐹 ∈ ( 𝑡m 𝑠) ∧ ∀𝑥𝑡 (𝐹𝑥) ∈ 𝑠))
1914, 18bitri 275 1 (𝐹 ran MblFnM ↔ ∃𝑠 ran sigAlgebra∃𝑡 ran sigAlgebra(𝐹 ∈ ( 𝑡m 𝑠) ∧ ∀𝑥𝑡 (𝐹𝑥) ∈ 𝑠))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1540  wcel 2108  wral 3061  wrex 3070  {crab 3436  cop 4632   cuni 4907   × cxp 5683  ccnv 5684  dom cdm 5685  ran crn 5686  cima 5688  Fun wfun 6555  cfv 6561  (class class class)co 7431  m cmap 8866  sigAlgebracsiga 34109  MblFnMcmbfm 34250
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-mbfm 34251
This theorem is referenced by:  mbfmfun  34254
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