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Theorem elunirnmbfm 34233
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 34231 . . . . 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 5345 . . . . 5 {𝑓 ∈ ( 𝑡m 𝑠) ∣ ∀𝑥𝑡 (𝑓𝑥) ∈ 𝑠} ∈ V
71, 6dmmpo 8095 . . . 4 dom MblFnM = ( ran sigAlgebra × ran sigAlgebra)
87rexeqi 3323 . . 3 (∃𝑎 ∈ dom MblFnM𝐹 ∈ (MblFnM‘𝑎) ↔ ∃𝑎 ∈ ( ran sigAlgebra × ran sigAlgebra)𝐹 ∈ (MblFnM‘𝑎))
9 fveq2 6907 . . . . . 6 (𝑎 = ⟨𝑠, 𝑡⟩ → (MblFnM‘𝑎) = (MblFnM‘⟨𝑠, 𝑡⟩))
10 df-ov 7434 . . . . . 6 (𝑠MblFnM𝑡) = (MblFnM‘⟨𝑠, 𝑡⟩)
119, 10eqtr4di 2793 . . . . 5 (𝑎 = ⟨𝑠, 𝑡⟩ → (MblFnM‘𝑎) = (𝑠MblFnM𝑡))
1211eleq2d 2825 . . . 4 (𝑎 = ⟨𝑠, 𝑡⟩ → (𝐹 ∈ (MblFnM‘𝑎) ↔ 𝐹 ∈ (𝑠MblFnM𝑡)))
1312rexxp 5856 . . 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 34232 . . 3 ((𝑠 ran sigAlgebra ∧ 𝑡 ran sigAlgebra) → (𝐹 ∈ (𝑠MblFnM𝑡) ↔ (𝐹 ∈ ( 𝑡m 𝑠) ∧ ∀𝑥𝑡 (𝐹𝑥) ∈ 𝑠)))
18172rexbiia 3216 . 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 1537  wcel 2106  wral 3059  wrex 3068  {crab 3433  cop 4637   cuni 4912   × cxp 5687  ccnv 5688  dom cdm 5689  ran crn 5690  cima 5692  Fun wfun 6557  cfv 6563  (class class class)co 7431  m cmap 8865  sigAlgebracsiga 34089  MblFnMcmbfm 34230
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-mbfm 34231
This theorem is referenced by:  mbfmfun  34234
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