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Theorem mbfmcnt 30560
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 30423 . . . . . 6 (𝑂𝑉 → 𝒫 𝑂 ∈ (sigAlgebra‘𝑂))
2 elrnsiga 30419 . . . . . 6 (𝒫 𝑂 ∈ (sigAlgebra‘𝑂) → 𝒫 𝑂 ran sigAlgebra)
31, 2syl 17 . . . . 5 (𝑂𝑉 → 𝒫 𝑂 ran sigAlgebra)
4 brsigarn 30477 . . . . . 6 𝔅 ∈ (sigAlgebra‘ℝ)
5 elrnsiga 30419 . . . . . 6 (𝔅 ∈ (sigAlgebra‘ℝ) → 𝔅 ran sigAlgebra)
64, 5mp1i 13 . . . . 5 (𝑂𝑉 → 𝔅 ran sigAlgebra)
73, 6ismbfm 30544 . . . 4 (𝑂𝑉 → (𝑓 ∈ (𝒫 𝑂MblFnM𝔅) ↔ (𝑓 ∈ ( 𝔅𝑚 𝒫 𝑂) ∧ ∀𝑥 ∈ 𝔅 (𝑓𝑥) ∈ 𝒫 𝑂)))
8 unibrsiga 30479 . . . . . . . . . 10 𝔅 = ℝ
9 reex 10140 . . . . . . . . . 10 ℝ ∈ V
108, 9eqeltri 2799 . . . . . . . . 9 𝔅 ∈ V
11 unipw 5023 . . . . . . . . . 10 𝒫 𝑂 = 𝑂
12 elex 3316 . . . . . . . . . 10 (𝑂𝑉𝑂 ∈ V)
1311, 12syl5eqel 2807 . . . . . . . . 9 (𝑂𝑉 𝒫 𝑂 ∈ V)
14 elmapg 7987 . . . . . . . . 9 (( 𝔅 ∈ V ∧ 𝒫 𝑂 ∈ V) → (𝑓 ∈ ( 𝔅𝑚 𝒫 𝑂) ↔ 𝑓: 𝒫 𝑂 𝔅))
1510, 13, 14sylancr 698 . . . . . . . 8 (𝑂𝑉 → (𝑓 ∈ ( 𝔅𝑚 𝒫 𝑂) ↔ 𝑓: 𝒫 𝑂 𝔅))
1611feq2i 6150 . . . . . . . 8 (𝑓: 𝒫 𝑂 𝔅𝑓:𝑂 𝔅)
1715, 16syl6bb 276 . . . . . . 7 (𝑂𝑉 → (𝑓 ∈ ( 𝔅𝑚 𝒫 𝑂) ↔ 𝑓:𝑂 𝔅))
18 ffn 6158 . . . . . . 7 (𝑓:𝑂 𝔅𝑓 Fn 𝑂)
1917, 18syl6bi 243 . . . . . 6 (𝑂𝑉 → (𝑓 ∈ ( 𝔅𝑚 𝒫 𝑂) → 𝑓 Fn 𝑂))
20 elpreima 6452 . . . . . . . . . 10 (𝑓 Fn 𝑂 → (𝑦 ∈ (𝑓𝑥) ↔ (𝑦𝑂 ∧ (𝑓𝑦) ∈ 𝑥)))
21 simpl 474 . . . . . . . . . 10 ((𝑦𝑂 ∧ (𝑓𝑦) ∈ 𝑥) → 𝑦𝑂)
2220, 21syl6bi 243 . . . . . . . . 9 (𝑓 Fn 𝑂 → (𝑦 ∈ (𝑓𝑥) → 𝑦𝑂))
2322ssrdv 3715 . . . . . . . 8 (𝑓 Fn 𝑂 → (𝑓𝑥) ⊆ 𝑂)
24 vex 3307 . . . . . . . . . . 11 𝑓 ∈ V
2524cnvex 7230 . . . . . . . . . 10 𝑓 ∈ V
26 imaexg 7220 . . . . . . . . . 10 (𝑓 ∈ V → (𝑓𝑥) ∈ V)
2725, 26ax-mp 5 . . . . . . . . 9 (𝑓𝑥) ∈ V
2827elpw 4272 . . . . . . . 8 ((𝑓𝑥) ∈ 𝒫 𝑂 ↔ (𝑓𝑥) ⊆ 𝑂)
2923, 28sylibr 224 . . . . . . 7 (𝑓 Fn 𝑂 → (𝑓𝑥) ∈ 𝒫 𝑂)
3029ralrimivw 3069 . . . . . 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 669 . . . 4 (𝑂𝑉 → (𝑓 ∈ ( 𝔅𝑚 𝒫 𝑂) ↔ (𝑓 ∈ ( 𝔅𝑚 𝒫 𝑂) ∧ ∀𝑥 ∈ 𝔅 (𝑓𝑥) ∈ 𝒫 𝑂)))
337, 32bitr4d 271 . . 3 (𝑂𝑉 → (𝑓 ∈ (𝒫 𝑂MblFnM𝔅) ↔ 𝑓 ∈ ( 𝔅𝑚 𝒫 𝑂)))
3433eqrdv 2722 . 2 (𝑂𝑉 → (𝒫 𝑂MblFnM𝔅) = ( 𝔅𝑚 𝒫 𝑂))
358, 11oveq12i 6777 . 2 ( 𝔅𝑚 𝒫 𝑂) = (ℝ ↑𝑚 𝑂)
3634, 35syl6eq 2774 1 (𝑂𝑉 → (𝒫 𝑂MblFnM𝔅) = (ℝ ↑𝑚 𝑂))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1596  wcel 2103  wral 3014  Vcvv 3304  wss 3680  𝒫 cpw 4266   cuni 4544  ccnv 5217  ran crn 5219  cima 5221   Fn wfn 5996  wf 5997  cfv 6001  (class class class)co 6765  𝑚 cmap 7974  cr 10048  sigAlgebracsiga 30400  𝔅cbrsiga 30474  MblFnMcmbfm 30542
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066  ax-cnex 10105  ax-resscn 10106  ax-pre-lttri 10123  ax-pre-lttrn 10124
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1599  df-fal 1602  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-nel 3000  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-int 4584  df-iun 4630  df-br 4761  df-opab 4821  df-mpt 4838  df-id 5128  df-po 5139  df-so 5140  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-ov 6768  df-oprab 6769  df-mpt2 6770  df-1st 7285  df-2nd 7286  df-er 7862  df-map 7976  df-en 8073  df-dom 8074  df-sdom 8075  df-pnf 10189  df-mnf 10190  df-xr 10191  df-ltxr 10192  df-le 10193  df-ioo 12293  df-topgen 16227  df-top 20822  df-bases 20873  df-siga 30401  df-sigagen 30432  df-brsiga 30475  df-mbfm 30543
This theorem is referenced by: (None)
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