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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)
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