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Theorem brae 29016
Description: 'almost everywhere' relation for a measure and a measurable set  A. (Contributed by Thierry Arnoux, 20-Oct-2017.)
Assertion
Ref Expression
brae  |-  ( ( M  e.  U. ran measures  /\  A  e.  dom  M )  ->  ( Aa.e. M  <->  ( M `  ( U. dom  M  \  A ) )  =  0 ) )

Proof of Theorem brae
Dummy variables  m  a are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 462 . . . . 5  |-  ( ( a  =  A  /\  m  =  M )  ->  m  =  M )
21dmeqd 4999 . . . . . . 7  |-  ( ( a  =  A  /\  m  =  M )  ->  dom  m  =  dom  M )
32unieqd 4172 . . . . . 6  |-  ( ( a  =  A  /\  m  =  M )  ->  U. dom  m  = 
U. dom  M )
4 simpl 458 . . . . . 6  |-  ( ( a  =  A  /\  m  =  M )  ->  a  =  A )
53, 4difeq12d 3527 . . . . 5  |-  ( ( a  =  A  /\  m  =  M )  ->  ( U. dom  m  \  a )  =  ( U. dom  M  \  A ) )
61, 5fveq12d 5831 . . . 4  |-  ( ( a  =  A  /\  m  =  M )  ->  ( m `  ( U. dom  m  \  a
) )  =  ( M `  ( U. dom  M  \  A ) ) )
76eqeq1d 2430 . . 3  |-  ( ( a  =  A  /\  m  =  M )  ->  ( ( m `  ( U. dom  m  \ 
a ) )  =  0  <->  ( M `  ( U. dom  M  \  A ) )  =  0 ) )
8 df-ae 29014 . . 3  |- a.e.  =  { <. a ,  m >.  |  ( m `  ( U. dom  m  \  a
) )  =  0 }
97, 8brabga 4677 . 2  |-  ( ( A  e.  dom  M  /\  M  e.  U. ran measures )  ->  ( Aa.e. M  <->  ( M `  ( U. dom  M  \  A ) )  =  0 ) )
109ancoms 454 1  |-  ( ( M  e.  U. ran measures  /\  A  e.  dom  M )  ->  ( Aa.e. M  <->  ( M `  ( U. dom  M  \  A ) )  =  0 ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1872    \ cdif 3376   U.cuni 4162   class class class wbr 4366   dom cdm 4796   ran crn 4797   ` cfv 5544   0cc0 9490  measurescmeas 28969  a.e.cae 29012
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-sep 4489  ax-nul 4498  ax-pr 4603
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rex 2720  df-rab 2723  df-v 3024  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-nul 3705  df-if 3855  df-sn 3942  df-pr 3944  df-op 3948  df-uni 4163  df-br 4367  df-opab 4426  df-dm 4806  df-iota 5508  df-fv 5552  df-ae 29014
This theorem is referenced by: (None)
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