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Theorem mbfmbfm 30865
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 30810 . . . 4 (𝑀 ran measures ↔ 𝑀 ∈ (measures‘dom 𝑀))
32biimpi 208 . . 3 (𝑀 ran measures → 𝑀 ∈ (measures‘dom 𝑀))
4 measbase 30805 . . 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 30750 . 2 (𝜑 → (sigaGen‘𝐽) ∈ ran sigAlgebra)
8 mbfmbfm.3 . 2 (𝜑𝐹 ∈ (dom 𝑀MblFnM(sigaGen‘𝐽)))
95, 7, 8isanmbfm 30863 1 (𝜑𝐹 ran MblFnM)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2166   cuni 4658  dom cdm 5342  ran crn 5343  cfv 6123  (class class class)co 6905  Topctop 21068  sigAlgebracsiga 30715  sigaGencsigagen 30746  measurescmeas 30803  MblFnMcmbfm 30857
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-fal 1672  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-int 4698  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-fv 6131  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-1st 7428  df-2nd 7429  df-esum 30635  df-siga 30716  df-sigagen 30747  df-meas 30804  df-mbfm 30858
This theorem is referenced by: (None)
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