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Theorem mbfmbfm 30654
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 30599 . . . 4 (𝑀 ran measures ↔ 𝑀 ∈ (measures‘dom 𝑀))
32biimpi 206 . . 3 (𝑀 ran measures → 𝑀 ∈ (measures‘dom 𝑀))
4 measbase 30594 . . 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 30539 . 2 (𝜑 → (sigaGen‘𝐽) ∈ ran sigAlgebra)
8 mbfmbfm.3 . 2 (𝜑𝐹 ∈ (dom 𝑀MblFnM(sigaGen‘𝐽)))
95, 7, 8isanmbfm 30652 1 (𝜑𝐹 ran MblFnM)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2144   cuni 4572  dom cdm 5249  ran crn 5250  cfv 6031  (class class class)co 6792  Topctop 20917  sigAlgebracsiga 30504  sigaGencsigagen 30535  measurescmeas 30592  MblFnMcmbfm 30646
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-fal 1636  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-int 4610  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-1st 7314  df-2nd 7315  df-esum 30424  df-siga 30505  df-sigagen 30536  df-meas 30593  df-mbfm 30647
This theorem is referenced by: (None)
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