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Theorem elunirnmbfm 30649
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 30647 . . . . 5 MblFnM = (𝑠 ran sigAlgebra, 𝑡 ran sigAlgebra ↦ {𝑓 ∈ ( 𝑡𝑚 𝑠) ∣ ∀𝑥𝑡 (𝑓𝑥) ∈ 𝑠})
21mpt2fun 6908 . . . 4 Fun MblFnM
3 elunirn 6651 . . . 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 6822 . . . . . 6 ( 𝑡𝑚 𝑠) ∈ V
65rabex 4943 . . . . 5 {𝑓 ∈ ( 𝑡𝑚 𝑠) ∣ ∀𝑥𝑡 (𝑓𝑥) ∈ 𝑠} ∈ V
71, 6dmmpt2 7389 . . . 4 dom MblFnM = ( ran sigAlgebra × ran sigAlgebra)
87rexeqi 3291 . . 3 (∃𝑎 ∈ dom MblFnM𝐹 ∈ (MblFnM‘𝑎) ↔ ∃𝑎 ∈ ( ran sigAlgebra × ran sigAlgebra)𝐹 ∈ (MblFnM‘𝑎))
9 fveq2 6332 . . . . . 6 (𝑎 = ⟨𝑠, 𝑡⟩ → (MblFnM‘𝑎) = (MblFnM‘⟨𝑠, 𝑡⟩))
10 df-ov 6795 . . . . . 6 (𝑠MblFnM𝑡) = (MblFnM‘⟨𝑠, 𝑡⟩)
119, 10syl6eqr 2822 . . . . 5 (𝑎 = ⟨𝑠, 𝑡⟩ → (MblFnM‘𝑎) = (𝑠MblFnM𝑡))
1211eleq2d 2835 . . . 4 (𝑎 = ⟨𝑠, 𝑡⟩ → (𝐹 ∈ (MblFnM‘𝑎) ↔ 𝐹 ∈ (𝑠MblFnM𝑡)))
1312rexxp 5403 . . 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 468 . . . 4 ((𝑠 ran sigAlgebra ∧ 𝑡 ran sigAlgebra) → 𝑠 ran sigAlgebra)
16 simpr 471 . . . 4 ((𝑠 ran sigAlgebra ∧ 𝑡 ran sigAlgebra) → 𝑡 ran sigAlgebra)
1715, 16ismbfm 30648 . . 3 ((𝑠 ran sigAlgebra ∧ 𝑡 ran sigAlgebra) → (𝐹 ∈ (𝑠MblFnM𝑡) ↔ (𝐹 ∈ ( 𝑡𝑚 𝑠) ∧ ∀𝑥𝑡 (𝐹𝑥) ∈ 𝑠)))
18172rexbiia 3202 . 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 382   = wceq 1630  wcel 2144  wral 3060  wrex 3061  {crab 3064  cop 4320   cuni 4572   × cxp 5247  ccnv 5248  dom cdm 5249  ran crn 5250  cima 5252  Fun wfun 6025  cfv 6031  (class class class)co 6792  𝑚 cmap 8008  sigAlgebracsiga 30504  MblFnMcmbfm 30646
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-1st 7314  df-2nd 7315  df-mbfm 30647
This theorem is referenced by:  mbfmfun  30650  isanmbfm  30652
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