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Theorem mbfmcnt 31425
Description: All functions are measurable with respect to the counting measure. (Contributed by Thierry Arnoux, 24-Jan-2017.)
Assertion
Ref Expression
mbfmcnt (𝑂𝑉 → (𝒫 𝑂MblFnM𝔅) = (ℝ ↑m 𝑂))

Proof of Theorem mbfmcnt
Dummy variables 𝑥 𝑓 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pwsiga 31288 . . . . . 6 (𝑂𝑉 → 𝒫 𝑂 ∈ (sigAlgebra‘𝑂))
2 elrnsiga 31284 . . . . . 6 (𝒫 𝑂 ∈ (sigAlgebra‘𝑂) → 𝒫 𝑂 ran sigAlgebra)
31, 2syl 17 . . . . 5 (𝑂𝑉 → 𝒫 𝑂 ran sigAlgebra)
4 brsigarn 31342 . . . . . 6 𝔅 ∈ (sigAlgebra‘ℝ)
5 elrnsiga 31284 . . . . . 6 (𝔅 ∈ (sigAlgebra‘ℝ) → 𝔅 ran sigAlgebra)
64, 5mp1i 13 . . . . 5 (𝑂𝑉 → 𝔅 ran sigAlgebra)
73, 6ismbfm 31409 . . . 4 (𝑂𝑉 → (𝑓 ∈ (𝒫 𝑂MblFnM𝔅) ↔ (𝑓 ∈ ( 𝔅m 𝒫 𝑂) ∧ ∀𝑥 ∈ 𝔅 (𝑓𝑥) ∈ 𝒫 𝑂)))
8 unibrsiga 31344 . . . . . . . . . 10 𝔅 = ℝ
9 reex 10616 . . . . . . . . . 10 ℝ ∈ V
108, 9eqeltri 2906 . . . . . . . . 9 𝔅 ∈ V
11 unipw 5333 . . . . . . . . . 10 𝒫 𝑂 = 𝑂
12 elex 3510 . . . . . . . . . 10 (𝑂𝑉𝑂 ∈ V)
1311, 12eqeltrid 2914 . . . . . . . . 9 (𝑂𝑉 𝒫 𝑂 ∈ V)
14 elmapg 8408 . . . . . . . . 9 (( 𝔅 ∈ V ∧ 𝒫 𝑂 ∈ V) → (𝑓 ∈ ( 𝔅m 𝒫 𝑂) ↔ 𝑓: 𝒫 𝑂 𝔅))
1510, 13, 14sylancr 587 . . . . . . . 8 (𝑂𝑉 → (𝑓 ∈ ( 𝔅m 𝒫 𝑂) ↔ 𝑓: 𝒫 𝑂 𝔅))
1611feq2i 6499 . . . . . . . 8 (𝑓: 𝒫 𝑂 𝔅𝑓:𝑂 𝔅)
1715, 16syl6bb 288 . . . . . . 7 (𝑂𝑉 → (𝑓 ∈ ( 𝔅m 𝒫 𝑂) ↔ 𝑓:𝑂 𝔅))
18 ffn 6507 . . . . . . 7 (𝑓:𝑂 𝔅𝑓 Fn 𝑂)
1917, 18syl6bi 254 . . . . . 6 (𝑂𝑉 → (𝑓 ∈ ( 𝔅m 𝒫 𝑂) → 𝑓 Fn 𝑂))
20 elpreima 6820 . . . . . . . . . 10 (𝑓 Fn 𝑂 → (𝑦 ∈ (𝑓𝑥) ↔ (𝑦𝑂 ∧ (𝑓𝑦) ∈ 𝑥)))
21 simpl 483 . . . . . . . . . 10 ((𝑦𝑂 ∧ (𝑓𝑦) ∈ 𝑥) → 𝑦𝑂)
2220, 21syl6bi 254 . . . . . . . . 9 (𝑓 Fn 𝑂 → (𝑦 ∈ (𝑓𝑥) → 𝑦𝑂))
2322ssrdv 3970 . . . . . . . 8 (𝑓 Fn 𝑂 → (𝑓𝑥) ⊆ 𝑂)
24 vex 3495 . . . . . . . . . . 11 𝑓 ∈ V
2524cnvex 7619 . . . . . . . . . 10 𝑓 ∈ V
26 imaexg 7609 . . . . . . . . . 10 (𝑓 ∈ V → (𝑓𝑥) ∈ V)
2725, 26ax-mp 5 . . . . . . . . 9 (𝑓𝑥) ∈ V
2827elpw 4542 . . . . . . . 8 ((𝑓𝑥) ∈ 𝒫 𝑂 ↔ (𝑓𝑥) ⊆ 𝑂)
2923, 28sylibr 235 . . . . . . 7 (𝑓 Fn 𝑂 → (𝑓𝑥) ∈ 𝒫 𝑂)
3029ralrimivw 3180 . . . . . 6 (𝑓 Fn 𝑂 → ∀𝑥 ∈ 𝔅 (𝑓𝑥) ∈ 𝒫 𝑂)
3119, 30syl6 35 . . . . 5 (𝑂𝑉 → (𝑓 ∈ ( 𝔅m 𝒫 𝑂) → ∀𝑥 ∈ 𝔅 (𝑓𝑥) ∈ 𝒫 𝑂))
3231pm4.71d 562 . . . 4 (𝑂𝑉 → (𝑓 ∈ ( 𝔅m 𝒫 𝑂) ↔ (𝑓 ∈ ( 𝔅m 𝒫 𝑂) ∧ ∀𝑥 ∈ 𝔅 (𝑓𝑥) ∈ 𝒫 𝑂)))
337, 32bitr4d 283 . . 3 (𝑂𝑉 → (𝑓 ∈ (𝒫 𝑂MblFnM𝔅) ↔ 𝑓 ∈ ( 𝔅m 𝒫 𝑂)))
3433eqrdv 2816 . 2 (𝑂𝑉 → (𝒫 𝑂MblFnM𝔅) = ( 𝔅m 𝒫 𝑂))
358, 11oveq12i 7157 . 2 ( 𝔅m 𝒫 𝑂) = (ℝ ↑m 𝑂)
3634, 35syl6eq 2869 1 (𝑂𝑉 → (𝒫 𝑂MblFnM𝔅) = (ℝ ↑m 𝑂))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wral 3135  Vcvv 3492  wss 3933  𝒫 cpw 4535   cuni 4830  ccnv 5547  ran crn 5549  cima 5551   Fn wfn 6343  wf 6344  cfv 6348  (class class class)co 7145  m cmap 8395  cr 10524  sigAlgebracsiga 31266  𝔅cbrsiga 31339  MblFnMcmbfm 31407
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-pre-lttri 10599  ax-pre-lttrn 10600
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-po 5467  df-so 5468  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7678  df-2nd 7679  df-er 8278  df-map 8397  df-en 8498  df-dom 8499  df-sdom 8500  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-ioo 12730  df-topgen 16705  df-top 21430  df-bases 21482  df-siga 31267  df-sigagen 31297  df-brsiga 31340  df-mbfm 31408
This theorem is referenced by: (None)
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