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Theorem mbfmbfm 26809
Description: A measurable function to a Borel Set is measurable. (Contributed by Thierry Arnoux, 24-Jan-2017.)
Hypotheses
Ref Expression
mbfmbfm.1  |-  ( ph  ->  M  e.  U. ran measures )
mbfmbfm.2  |-  ( ph  ->  J  e.  Top )
mbfmbfm.3  |-  ( ph  ->  F  e.  ( dom 
MMblFnM (sigaGen `  J )
) )
Assertion
Ref Expression
mbfmbfm  |-  ( ph  ->  F  e.  U. ran MblFnM )

Proof of Theorem mbfmbfm
StepHypRef Expression
1 mbfmbfm.1 . . 3  |-  ( ph  ->  M  e.  U. ran measures )
2 measbasedom 26752 . . . 4  |-  ( M  e.  U. ran measures  <->  M  e.  (measures `  dom  M ) )
32biimpi 194 . . 3  |-  ( M  e.  U. ran measures  ->  M  e.  (measures `  dom  M ) )
4 measbase 26747 . . 3  |-  ( M  e.  (measures `  dom  M )  ->  dom  M  e. 
U. ran sigAlgebra )
51, 3, 43syl 20 . 2  |-  ( ph  ->  dom  M  e.  U. ran sigAlgebra )
6 mbfmbfm.2 . . 3  |-  ( ph  ->  J  e.  Top )
76sgsiga 26721 . 2  |-  ( ph  ->  (sigaGen `  J )  e.  U. ran sigAlgebra )
8 mbfmbfm.3 . 2  |-  ( ph  ->  F  e.  ( dom 
MMblFnM (sigaGen `  J )
) )
95, 7, 8isanmbfm 26807 1  |-  ( ph  ->  F  e.  U. ran MblFnM )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1758   U.cuni 4191   dom cdm 4940   ran crn 4941   ` cfv 5518  (class class class)co 6192   Topctop 18616  sigAlgebracsiga 26686  sigaGencsigagen 26717  measurescmeas 26745  MblFnMcmbfm 26801
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-int 4229  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-fv 5526  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-1st 6679  df-2nd 6680  df-esum 26620  df-siga 26687  df-sigagen 26718  df-meas 26746  df-mbfm 26802
This theorem is referenced by: (None)
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