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Theorem mbfmcnt 30129
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 29992 . . . . . 6 (𝑂𝑉 → 𝒫 𝑂 ∈ (sigAlgebra‘𝑂))
2 elrnsiga 29988 . . . . . 6 (𝒫 𝑂 ∈ (sigAlgebra‘𝑂) → 𝒫 𝑂 ran sigAlgebra)
31, 2syl 17 . . . . 5 (𝑂𝑉 → 𝒫 𝑂 ran sigAlgebra)
4 brsigarn 30046 . . . . . 6 𝔅 ∈ (sigAlgebra‘ℝ)
5 elrnsiga 29988 . . . . . 6 (𝔅 ∈ (sigAlgebra‘ℝ) → 𝔅 ran sigAlgebra)
64, 5mp1i 13 . . . . 5 (𝑂𝑉 → 𝔅 ran sigAlgebra)
73, 6ismbfm 30113 . . . 4 (𝑂𝑉 → (𝑓 ∈ (𝒫 𝑂MblFnM𝔅) ↔ (𝑓 ∈ ( 𝔅𝑚 𝒫 𝑂) ∧ ∀𝑥 ∈ 𝔅 (𝑓𝑥) ∈ 𝒫 𝑂)))
8 unibrsiga 30048 . . . . . . . . . 10 𝔅 = ℝ
9 reex 9978 . . . . . . . . . 10 ℝ ∈ V
108, 9eqeltri 2694 . . . . . . . . 9 𝔅 ∈ V
11 unipw 4884 . . . . . . . . . 10 𝒫 𝑂 = 𝑂
12 elex 3201 . . . . . . . . . 10 (𝑂𝑉𝑂 ∈ V)
1311, 12syl5eqel 2702 . . . . . . . . 9 (𝑂𝑉 𝒫 𝑂 ∈ V)
14 elmapg 7822 . . . . . . . . 9 (( 𝔅 ∈ V ∧ 𝒫 𝑂 ∈ V) → (𝑓 ∈ ( 𝔅𝑚 𝒫 𝑂) ↔ 𝑓: 𝒫 𝑂 𝔅))
1510, 13, 14sylancr 694 . . . . . . . 8 (𝑂𝑉 → (𝑓 ∈ ( 𝔅𝑚 𝒫 𝑂) ↔ 𝑓: 𝒫 𝑂 𝔅))
1611feq2i 5999 . . . . . . . 8 (𝑓: 𝒫 𝑂 𝔅𝑓:𝑂 𝔅)
1715, 16syl6bb 276 . . . . . . 7 (𝑂𝑉 → (𝑓 ∈ ( 𝔅𝑚 𝒫 𝑂) ↔ 𝑓:𝑂 𝔅))
18 ffn 6007 . . . . . . 7 (𝑓:𝑂 𝔅𝑓 Fn 𝑂)
1917, 18syl6bi 243 . . . . . 6 (𝑂𝑉 → (𝑓 ∈ ( 𝔅𝑚 𝒫 𝑂) → 𝑓 Fn 𝑂))
20 elpreima 6298 . . . . . . . . . 10 (𝑓 Fn 𝑂 → (𝑦 ∈ (𝑓𝑥) ↔ (𝑦𝑂 ∧ (𝑓𝑦) ∈ 𝑥)))
21 simpl 473 . . . . . . . . . 10 ((𝑦𝑂 ∧ (𝑓𝑦) ∈ 𝑥) → 𝑦𝑂)
2220, 21syl6bi 243 . . . . . . . . 9 (𝑓 Fn 𝑂 → (𝑦 ∈ (𝑓𝑥) → 𝑦𝑂))
2322ssrdv 3593 . . . . . . . 8 (𝑓 Fn 𝑂 → (𝑓𝑥) ⊆ 𝑂)
24 vex 3192 . . . . . . . . . . 11 𝑓 ∈ V
2524cnvex 7067 . . . . . . . . . 10 𝑓 ∈ V
26 imaexg 7057 . . . . . . . . . 10 (𝑓 ∈ V → (𝑓𝑥) ∈ V)
2725, 26ax-mp 5 . . . . . . . . 9 (𝑓𝑥) ∈ V
2827elpw 4141 . . . . . . . 8 ((𝑓𝑥) ∈ 𝒫 𝑂 ↔ (𝑓𝑥) ⊆ 𝑂)
2923, 28sylibr 224 . . . . . . 7 (𝑓 Fn 𝑂 → (𝑓𝑥) ∈ 𝒫 𝑂)
3029ralrimivw 2962 . . . . . 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 665 . . . 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 2619 . 2 (𝑂𝑉 → (𝒫 𝑂MblFnM𝔅) = ( 𝔅𝑚 𝒫 𝑂))
358, 11oveq12i 6622 . 2 ( 𝔅𝑚 𝒫 𝑂) = (ℝ ↑𝑚 𝑂)
3634, 35syl6eq 2671 1 (𝑂𝑉 → (𝒫 𝑂MblFnM𝔅) = (ℝ ↑𝑚 𝑂))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2907  Vcvv 3189  wss 3559  𝒫 cpw 4135   cuni 4407  ccnv 5078  ran crn 5080  cima 5082   Fn wfn 5847  wf 5848  cfv 5852  (class class class)co 6610  𝑚 cmap 7809  cr 9886  sigAlgebracsiga 29969  𝔅cbrsiga 30043  MblFnMcmbfm 30111
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 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-cnex 9943  ax-resscn 9944  ax-pre-lttri 9961  ax-pre-lttrn 9962
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  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-nel 2894  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-po 5000  df-so 5001  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-1st 7120  df-2nd 7121  df-er 7694  df-map 7811  df-en 7907  df-dom 7908  df-sdom 7909  df-pnf 10027  df-mnf 10028  df-xr 10029  df-ltxr 10030  df-le 10031  df-ioo 12128  df-topgen 16032  df-top 20627  df-bases 20670  df-siga 29970  df-sigagen 30001  df-brsiga 30044  df-mbfm 30112
This theorem is referenced by: (None)
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