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Theorem elunirnmbfm 30138
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 30136 . . . . 5 MblFnM = (𝑠 ran sigAlgebra, 𝑡 ran sigAlgebra ↦ {𝑓 ∈ ( 𝑡𝑚 𝑠) ∣ ∀𝑥𝑡 (𝑓𝑥) ∈ 𝑠})
21mpt2fun 6727 . . . 4 Fun MblFnM
3 elunirn 6474 . . . 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 6643 . . . . . 6 ( 𝑡𝑚 𝑠) ∈ V
65rabex 4783 . . . . 5 {𝑓 ∈ ( 𝑡𝑚 𝑠) ∣ ∀𝑥𝑡 (𝑓𝑥) ∈ 𝑠} ∈ V
71, 6dmmpt2 7200 . . . 4 dom MblFnM = ( ran sigAlgebra × ran sigAlgebra)
87rexeqi 3136 . . 3 (∃𝑎 ∈ dom MblFnM𝐹 ∈ (MblFnM‘𝑎) ↔ ∃𝑎 ∈ ( ran sigAlgebra × ran sigAlgebra)𝐹 ∈ (MblFnM‘𝑎))
9 fveq2 6158 . . . . . 6 (𝑎 = ⟨𝑠, 𝑡⟩ → (MblFnM‘𝑎) = (MblFnM‘⟨𝑠, 𝑡⟩))
10 df-ov 6618 . . . . . 6 (𝑠MblFnM𝑡) = (MblFnM‘⟨𝑠, 𝑡⟩)
119, 10syl6eqr 2673 . . . . 5 (𝑎 = ⟨𝑠, 𝑡⟩ → (MblFnM‘𝑎) = (𝑠MblFnM𝑡))
1211eleq2d 2684 . . . 4 (𝑎 = ⟨𝑠, 𝑡⟩ → (𝐹 ∈ (MblFnM‘𝑎) ↔ 𝐹 ∈ (𝑠MblFnM𝑡)))
1312rexxp 5234 . . 3 (∃𝑎 ∈ ( ran sigAlgebra × ran sigAlgebra)𝐹 ∈ (MblFnM‘𝑎) ↔ ∃𝑠 ran sigAlgebra∃𝑡 ran sigAlgebra𝐹 ∈ (𝑠MblFnM𝑡))
144, 8, 133bitri 286 . 2 (𝐹 ran MblFnM ↔ ∃𝑠 ran sigAlgebra∃𝑡 ran sigAlgebra𝐹 ∈ (𝑠MblFnM𝑡))
15 simpl 473 . . . 4 ((𝑠 ran sigAlgebra ∧ 𝑡 ran sigAlgebra) → 𝑠 ran sigAlgebra)
16 simpr 477 . . . 4 ((𝑠 ran sigAlgebra ∧ 𝑡 ran sigAlgebra) → 𝑡 ran sigAlgebra)
1715, 16ismbfm 30137 . . 3 ((𝑠 ran sigAlgebra ∧ 𝑡 ran sigAlgebra) → (𝐹 ∈ (𝑠MblFnM𝑡) ↔ (𝐹 ∈ ( 𝑡𝑚 𝑠) ∧ ∀𝑥𝑡 (𝐹𝑥) ∈ 𝑠)))
18172rexbiia 3050 . 2 (∃𝑠 ran sigAlgebra∃𝑡 ran sigAlgebra𝐹 ∈ (𝑠MblFnM𝑡) ↔ ∃𝑠 ran sigAlgebra∃𝑡 ran sigAlgebra(𝐹 ∈ ( 𝑡𝑚 𝑠) ∧ ∀𝑥𝑡 (𝐹𝑥) ∈ 𝑠))
1914, 18bitri 264 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 196  wa 384   = wceq 1480  wcel 1987  wral 2908  wrex 2909  {crab 2912  cop 4161   cuni 4409   × cxp 5082  ccnv 5083  dom cdm 5084  ran crn 5085  cima 5087  Fun wfun 5851  cfv 5857  (class class class)co 6615  𝑚 cmap 7817  sigAlgebracsiga 29993  MblFnMcmbfm 30135
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 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
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-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-1st 7128  df-2nd 7129  df-mbfm 30136
This theorem is referenced by:  mbfmfun  30139  isanmbfm  30141
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