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Theorem mbfmbfm 30294
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 30239 . . . 4 (𝑀 ran measures ↔ 𝑀 ∈ (measures‘dom 𝑀))
32biimpi 206 . . 3 (𝑀 ran measures → 𝑀 ∈ (measures‘dom 𝑀))
4 measbase 30234 . . 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 30179 . 2 (𝜑 → (sigaGen‘𝐽) ∈ ran sigAlgebra)
8 mbfmbfm.3 . 2 (𝜑𝐹 ∈ (dom 𝑀MblFnM(sigaGen‘𝐽)))
95, 7, 8isanmbfm 30292 1 (𝜑𝐹 ran MblFnM)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1988   cuni 4427  dom cdm 5104  ran crn 5105  cfv 5876  (class class class)co 6635  Topctop 20679  sigAlgebracsiga 30144  sigaGencsigagen 30175  measurescmeas 30232  MblFnMcmbfm 30286
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-fal 1487  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-1st 7153  df-2nd 7154  df-esum 30064  df-siga 30145  df-sigagen 30176  df-meas 30233  df-mbfm 30287
This theorem is referenced by: (None)
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