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Theorem mbfmcnt 30649
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 30512 . . . . . 6 (𝑂𝑉 → 𝒫 𝑂 ∈ (sigAlgebra‘𝑂))
2 elrnsiga 30508 . . . . . 6 (𝒫 𝑂 ∈ (sigAlgebra‘𝑂) → 𝒫 𝑂 ran sigAlgebra)
31, 2syl 17 . . . . 5 (𝑂𝑉 → 𝒫 𝑂 ran sigAlgebra)
4 brsigarn 30566 . . . . . 6 𝔅 ∈ (sigAlgebra‘ℝ)
5 elrnsiga 30508 . . . . . 6 (𝔅 ∈ (sigAlgebra‘ℝ) → 𝔅 ran sigAlgebra)
64, 5mp1i 13 . . . . 5 (𝑂𝑉 → 𝔅 ran sigAlgebra)
73, 6ismbfm 30633 . . . 4 (𝑂𝑉 → (𝑓 ∈ (𝒫 𝑂MblFnM𝔅) ↔ (𝑓 ∈ ( 𝔅𝑚 𝒫 𝑂) ∧ ∀𝑥 ∈ 𝔅 (𝑓𝑥) ∈ 𝒫 𝑂)))
8 unibrsiga 30568 . . . . . . . . . 10 𝔅 = ℝ
9 reex 10306 . . . . . . . . . 10 ℝ ∈ V
108, 9eqeltri 2877 . . . . . . . . 9 𝔅 ∈ V
11 unipw 5102 . . . . . . . . . 10 𝒫 𝑂 = 𝑂
12 elex 3402 . . . . . . . . . 10 (𝑂𝑉𝑂 ∈ V)
1311, 12syl5eqel 2885 . . . . . . . . 9 (𝑂𝑉 𝒫 𝑂 ∈ V)
14 elmapg 8099 . . . . . . . . 9 (( 𝔅 ∈ V ∧ 𝒫 𝑂 ∈ V) → (𝑓 ∈ ( 𝔅𝑚 𝒫 𝑂) ↔ 𝑓: 𝒫 𝑂 𝔅))
1510, 13, 14sylancr 577 . . . . . . . 8 (𝑂𝑉 → (𝑓 ∈ ( 𝔅𝑚 𝒫 𝑂) ↔ 𝑓: 𝒫 𝑂 𝔅))
1611feq2i 6242 . . . . . . . 8 (𝑓: 𝒫 𝑂 𝔅𝑓:𝑂 𝔅)
1715, 16syl6bb 278 . . . . . . 7 (𝑂𝑉 → (𝑓 ∈ ( 𝔅𝑚 𝒫 𝑂) ↔ 𝑓:𝑂 𝔅))
18 ffn 6250 . . . . . . 7 (𝑓:𝑂 𝔅𝑓 Fn 𝑂)
1917, 18syl6bi 244 . . . . . 6 (𝑂𝑉 → (𝑓 ∈ ( 𝔅𝑚 𝒫 𝑂) → 𝑓 Fn 𝑂))
20 elpreima 6553 . . . . . . . . . 10 (𝑓 Fn 𝑂 → (𝑦 ∈ (𝑓𝑥) ↔ (𝑦𝑂 ∧ (𝑓𝑦) ∈ 𝑥)))
21 simpl 470 . . . . . . . . . 10 ((𝑦𝑂 ∧ (𝑓𝑦) ∈ 𝑥) → 𝑦𝑂)
2220, 21syl6bi 244 . . . . . . . . 9 (𝑓 Fn 𝑂 → (𝑦 ∈ (𝑓𝑥) → 𝑦𝑂))
2322ssrdv 3798 . . . . . . . 8 (𝑓 Fn 𝑂 → (𝑓𝑥) ⊆ 𝑂)
24 vex 3390 . . . . . . . . . . 11 𝑓 ∈ V
2524cnvex 7337 . . . . . . . . . 10 𝑓 ∈ V
26 imaexg 7327 . . . . . . . . . 10 (𝑓 ∈ V → (𝑓𝑥) ∈ V)
2725, 26ax-mp 5 . . . . . . . . 9 (𝑓𝑥) ∈ V
2827elpw 4351 . . . . . . . 8 ((𝑓𝑥) ∈ 𝒫 𝑂 ↔ (𝑓𝑥) ⊆ 𝑂)
2923, 28sylibr 225 . . . . . . 7 (𝑓 Fn 𝑂 → (𝑓𝑥) ∈ 𝒫 𝑂)
3029ralrimivw 3151 . . . . . 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 553 . . . 4 (𝑂𝑉 → (𝑓 ∈ ( 𝔅𝑚 𝒫 𝑂) ↔ (𝑓 ∈ ( 𝔅𝑚 𝒫 𝑂) ∧ ∀𝑥 ∈ 𝔅 (𝑓𝑥) ∈ 𝒫 𝑂)))
337, 32bitr4d 273 . . 3 (𝑂𝑉 → (𝑓 ∈ (𝒫 𝑂MblFnM𝔅) ↔ 𝑓 ∈ ( 𝔅𝑚 𝒫 𝑂)))
3433eqrdv 2800 . 2 (𝑂𝑉 → (𝒫 𝑂MblFnM𝔅) = ( 𝔅𝑚 𝒫 𝑂))
358, 11oveq12i 6880 . 2 ( 𝔅𝑚 𝒫 𝑂) = (ℝ ↑𝑚 𝑂)
3634, 35syl6eq 2852 1 (𝑂𝑉 → (𝒫 𝑂MblFnM𝔅) = (ℝ ↑𝑚 𝑂))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1637  wcel 2155  wral 3092  Vcvv 3387  wss 3763  𝒫 cpw 4345   cuni 4623  ccnv 5304  ran crn 5306  cima 5308   Fn wfn 6090  wf 6091  cfv 6095  (class class class)co 6868  𝑚 cmap 8086  cr 10214  sigAlgebracsiga 30489  𝔅cbrsiga 30563  MblFnMcmbfm 30631
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2067  ax-7 2103  ax-8 2157  ax-9 2164  ax-10 2184  ax-11 2200  ax-12 2213  ax-13 2419  ax-ext 2781  ax-sep 4968  ax-nul 4977  ax-pow 5029  ax-pr 5090  ax-un 7173  ax-cnex 10271  ax-resscn 10272  ax-pre-lttri 10289  ax-pre-lttrn 10290
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-fal 1651  df-ex 1860  df-nf 1864  df-sb 2060  df-eu 2633  df-mo 2634  df-clab 2789  df-cleq 2795  df-clel 2798  df-nfc 2933  df-ne 2975  df-nel 3078  df-ral 3097  df-rex 3098  df-rab 3101  df-v 3389  df-sbc 3628  df-csb 3723  df-dif 3766  df-un 3768  df-in 3770  df-ss 3777  df-nul 4111  df-if 4274  df-pw 4347  df-sn 4365  df-pr 4367  df-op 4371  df-uni 4624  df-int 4663  df-iun 4707  df-br 4838  df-opab 4900  df-mpt 4917  df-id 5213  df-po 5226  df-so 5227  df-xp 5311  df-rel 5312  df-cnv 5313  df-co 5314  df-dm 5315  df-rn 5316  df-res 5317  df-ima 5318  df-iota 6058  df-fun 6097  df-fn 6098  df-f 6099  df-f1 6100  df-fo 6101  df-f1o 6102  df-fv 6103  df-ov 6871  df-oprab 6872  df-mpt2 6873  df-1st 7392  df-2nd 7393  df-er 7973  df-map 8088  df-en 8187  df-dom 8188  df-sdom 8189  df-pnf 10355  df-mnf 10356  df-xr 10357  df-ltxr 10358  df-le 10359  df-ioo 12391  df-topgen 16303  df-top 20906  df-bases 20958  df-siga 30490  df-sigagen 30521  df-brsiga 30564  df-mbfm 30632
This theorem is referenced by: (None)
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