Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  mbfmbfm Structured version   Visualization version   GIF version

Theorem mbfmbfm 31590
Description: A measurable function to a Borel Set is measurable. (Contributed by Thierry Arnoux, 24-Jan-2017.)
Hypotheses
Ref Expression
mbfmbfm.1 (𝜑𝑀 ran measures)
mbfmbfm.2 (𝜑𝐽 ∈ Top)
mbfmbfm.3 (𝜑𝐹 ∈ (dom 𝑀MblFnM(sigaGen‘𝐽)))
Assertion
Ref Expression
mbfmbfm (𝜑𝐹 ran MblFnM)

Proof of Theorem mbfmbfm
StepHypRef Expression
1 mbfmbfm.1 . . 3 (𝜑𝑀 ran measures)
2 measbasedom 31535 . . . 4 (𝑀 ran measures ↔ 𝑀 ∈ (measures‘dom 𝑀))
32biimpi 219 . . 3 (𝑀 ran measures → 𝑀 ∈ (measures‘dom 𝑀))
4 measbase 31530 . . 3 (𝑀 ∈ (measures‘dom 𝑀) → dom 𝑀 ran sigAlgebra)
51, 3, 43syl 18 . 2 (𝜑 → dom 𝑀 ran sigAlgebra)
6 mbfmbfm.2 . . 3 (𝜑𝐽 ∈ Top)
76sgsiga 31475 . 2 (𝜑 → (sigaGen‘𝐽) ∈ ran sigAlgebra)
8 mbfmbfm.3 . 2 (𝜑𝐹 ∈ (dom 𝑀MblFnM(sigaGen‘𝐽)))
95, 7, 8isanmbfm 31588 1 (𝜑𝐹 ran MblFnM)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2114   cuni 4813  dom cdm 5532  ran crn 5533  cfv 6334  (class class class)co 7140  Topctop 21496  sigAlgebracsiga 31441  sigaGencsigagen 31471  measurescmeas 31528  MblFnMcmbfm 31582
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-int 4852  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-fv 6342  df-ov 7143  df-oprab 7144  df-mpo 7145  df-1st 7675  df-2nd 7676  df-esum 31361  df-siga 31442  df-sigagen 31472  df-meas 31529  df-mbfm 31583
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator