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Theorem elunirnmbfm 31511
 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 31509 . . . . 5 MblFnM = (𝑠 ran sigAlgebra, 𝑡 ran sigAlgebra ↦ {𝑓 ∈ ( 𝑡m 𝑠) ∣ ∀𝑥𝑡 (𝑓𝑥) ∈ 𝑠})
21mpofun 7275 . . . 4 Fun MblFnM
3 elunirn 7009 . . . 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 7188 . . . . . 6 ( 𝑡m 𝑠) ∈ V
65rabex 5234 . . . . 5 {𝑓 ∈ ( 𝑡m 𝑠) ∣ ∀𝑥𝑡 (𝑓𝑥) ∈ 𝑠} ∈ V
71, 6dmmpo 7768 . . . 4 dom MblFnM = ( ran sigAlgebra × ran sigAlgebra)
87rexeqi 3414 . . 3 (∃𝑎 ∈ dom MblFnM𝐹 ∈ (MblFnM‘𝑎) ↔ ∃𝑎 ∈ ( ran sigAlgebra × ran sigAlgebra)𝐹 ∈ (MblFnM‘𝑎))
9 fveq2 6669 . . . . . 6 (𝑎 = ⟨𝑠, 𝑡⟩ → (MblFnM‘𝑎) = (MblFnM‘⟨𝑠, 𝑡⟩))
10 df-ov 7158 . . . . . 6 (𝑠MblFnM𝑡) = (MblFnM‘⟨𝑠, 𝑡⟩)
119, 10syl6eqr 2874 . . . . 5 (𝑎 = ⟨𝑠, 𝑡⟩ → (MblFnM‘𝑎) = (𝑠MblFnM𝑡))
1211eleq2d 2898 . . . 4 (𝑎 = ⟨𝑠, 𝑡⟩ → (𝐹 ∈ (MblFnM‘𝑎) ↔ 𝐹 ∈ (𝑠MblFnM𝑡)))
1312rexxp 5712 . . 3 (∃𝑎 ∈ ( ran sigAlgebra × ran sigAlgebra)𝐹 ∈ (MblFnM‘𝑎) ↔ ∃𝑠 ran sigAlgebra∃𝑡 ran sigAlgebra𝐹 ∈ (𝑠MblFnM𝑡))
144, 8, 133bitri 299 . 2 (𝐹 ran MblFnM ↔ ∃𝑠 ran sigAlgebra∃𝑡 ran sigAlgebra𝐹 ∈ (𝑠MblFnM𝑡))
15 simpl 485 . . . 4 ((𝑠 ran sigAlgebra ∧ 𝑡 ran sigAlgebra) → 𝑠 ran sigAlgebra)
16 simpr 487 . . . 4 ((𝑠 ran sigAlgebra ∧ 𝑡 ran sigAlgebra) → 𝑡 ran sigAlgebra)
1715, 16ismbfm 31510 . . 3 ((𝑠 ran sigAlgebra ∧ 𝑡 ran sigAlgebra) → (𝐹 ∈ (𝑠MblFnM𝑡) ↔ (𝐹 ∈ ( 𝑡m 𝑠) ∧ ∀𝑥𝑡 (𝐹𝑥) ∈ 𝑠)))
18172rexbiia 3298 . 2 (∃𝑠 ran sigAlgebra∃𝑡 ran sigAlgebra𝐹 ∈ (𝑠MblFnM𝑡) ↔ ∃𝑠 ran sigAlgebra∃𝑡 ran sigAlgebra(𝐹 ∈ ( 𝑡m 𝑠) ∧ ∀𝑥𝑡 (𝐹𝑥) ∈ 𝑠))
1914, 18bitri 277 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 208   ∧ wa 398   = wceq 1533   ∈ wcel 2110  ∀wral 3138  ∃wrex 3139  {crab 3142  ⟨cop 4572  ∪ cuni 4837   × cxp 5552  ◡ccnv 5553  dom cdm 5554  ran crn 5555   “ cima 5557  Fun wfun 6348  ‘cfv 6354  (class class class)co 7155   ↑m cmap 8405  sigAlgebracsiga 31367  MblFnMcmbfm 31508 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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-fv 6362  df-ov 7158  df-oprab 7159  df-mpo 7160  df-1st 7688  df-2nd 7689  df-mbfm 31509 This theorem is referenced by:  mbfmfun  31512  isanmbfm  31514
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