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Theorem brae 27869
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 461 . . . . 5  |-  ( ( a  =  A  /\  m  =  M )  ->  m  =  M )
21dmeqd 5204 . . . . . . 7  |-  ( ( a  =  A  /\  m  =  M )  ->  dom  m  =  dom  M )
32unieqd 4255 . . . . . 6  |-  ( ( a  =  A  /\  m  =  M )  ->  U. dom  m  = 
U. dom  M )
4 simpl 457 . . . . . 6  |-  ( ( a  =  A  /\  m  =  M )  ->  a  =  A )
53, 4difeq12d 3623 . . . . 5  |-  ( ( a  =  A  /\  m  =  M )  ->  ( U. dom  m  \  a )  =  ( U. dom  M  \  A ) )
61, 5fveq12d 5871 . . . 4  |-  ( ( a  =  A  /\  m  =  M )  ->  ( m `  ( U. dom  m  \  a
) )  =  ( M `  ( U. dom  M  \  A ) ) )
76eqeq1d 2469 . . 3  |-  ( ( a  =  A  /\  m  =  M )  ->  ( ( m `  ( U. dom  m  \ 
a ) )  =  0  <->  ( M `  ( U. dom  M  \  A ) )  =  0 ) )
8 df-ae 27867 . . 3  |- a.e.  =  { <. a ,  m >.  |  ( m `  ( U. dom  m  \  a
) )  =  0 }
97, 8brabga 4761 . 2  |-  ( ( A  e.  dom  M  /\  M  e.  U. ran measures )  ->  ( Aa.e. M  <->  ( M `  ( U. dom  M  \  A ) )  =  0 ) )
109ancoms 453 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 184    /\ wa 369    = wceq 1379    e. wcel 1767    \ cdif 3473   U.cuni 4245   class class class wbr 4447   dom cdm 4999   ran crn 5000   ` cfv 5587   0cc0 9491  measurescmeas 27822  a.e.cae 27865
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-dm 5009  df-iota 5550  df-fv 5595  df-ae 27867
This theorem is referenced by: (None)
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