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Theorem braew 26610
Description: 'almost everywhere' relation for a measure  M and a property  ph (Contributed by Thierry Arnoux, 20-Oct-2017.)
Hypothesis
Ref Expression
braew.1  |-  U. dom  M  =  O
Assertion
Ref Expression
braew  |-  ( M  e.  U. ran measures  ->  ( { x  e.  O  |  ph }a.e. M  <->  ( M `  { x  e.  O  |  -.  ph } )  =  0 ) )
Distinct variable group:    x, O
Allowed substitution hints:    ph( x)    M( x)

Proof of Theorem braew
Dummy variables  m  a are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 braew.1 . . . . 5  |-  U. dom  M  =  O
2 dmexg 6504 . . . . . 6  |-  ( M  e.  U. ran measures  ->  dom  M  e.  _V )
3 uniexg 6372 . . . . . 6  |-  ( dom 
M  e.  _V  ->  U.
dom  M  e.  _V )
42, 3syl 16 . . . . 5  |-  ( M  e.  U. ran measures  ->  U. dom  M  e.  _V )
51, 4syl5eqelr 2523 . . . 4  |-  ( M  e.  U. ran measures  ->  O  e.  _V )
6 rabexg 4437 . . . 4  |-  ( O  e.  _V  ->  { x  e.  O  |  ph }  e.  _V )
75, 6syl 16 . . 3  |-  ( M  e.  U. ran measures  ->  { x  e.  O  |  ph }  e.  _V )
8 simpr 461 . . . . . 6  |-  ( ( a  =  { x  e.  O  |  ph }  /\  m  =  M
)  ->  m  =  M )
98dmeqd 5037 . . . . . . . 8  |-  ( ( a  =  { x  e.  O  |  ph }  /\  m  =  M
)  ->  dom  m  =  dom  M )
109unieqd 4096 . . . . . . 7  |-  ( ( a  =  { x  e.  O  |  ph }  /\  m  =  M
)  ->  U. dom  m  =  U. dom  M )
11 simpl 457 . . . . . . 7  |-  ( ( a  =  { x  e.  O  |  ph }  /\  m  =  M
)  ->  a  =  { x  e.  O  |  ph } )
1210, 11difeq12d 3470 . . . . . 6  |-  ( ( a  =  { x  e.  O  |  ph }  /\  m  =  M
)  ->  ( U. dom  m  \  a )  =  ( U. dom  M 
\  { x  e.  O  |  ph }
) )
138, 12fveq12d 5692 . . . . 5  |-  ( ( a  =  { x  e.  O  |  ph }  /\  m  =  M
)  ->  ( m `  ( U. dom  m  \  a ) )  =  ( M `  ( U. dom  M  \  {
x  e.  O  |  ph } ) ) )
1413eqeq1d 2446 . . . 4  |-  ( ( a  =  { x  e.  O  |  ph }  /\  m  =  M
)  ->  ( (
m `  ( U. dom  m  \  a ) )  =  0  <->  ( M `  ( U. dom  M  \  { x  e.  O  |  ph }
) )  =  0 ) )
15 df-ae 26607 . . . 4  |- a.e.  =  { <. a ,  m >.  |  ( m `  ( U. dom  m  \  a
) )  =  0 }
1614, 15brabga 4598 . . 3  |-  ( ( { x  e.  O  |  ph }  e.  _V  /\  M  e.  U. ran measures )  ->  ( { x  e.  O  |  ph }a.e. M 
<->  ( M `  ( U. dom  M  \  {
x  e.  O  |  ph } ) )  =  0 ) )
177, 16mpancom 669 . 2  |-  ( M  e.  U. ran measures  ->  ( { x  e.  O  |  ph }a.e. M  <->  ( M `  ( U. dom  M  \  { x  e.  O  |  ph } ) )  =  0 ) )
181difeq1i 3465 . . . . 5  |-  ( U. dom  M  \  { x  e.  O  |  ph }
)  =  ( O 
\  { x  e.  O  |  ph }
)
19 notrab 3622 . . . . 5  |-  ( O 
\  { x  e.  O  |  ph }
)  =  { x  e.  O  |  -.  ph }
2018, 19eqtri 2458 . . . 4  |-  ( U. dom  M  \  { x  e.  O  |  ph }
)  =  { x  e.  O  |  -.  ph }
2120fveq2i 5689 . . 3  |-  ( M `
 ( U. dom  M 
\  { x  e.  O  |  ph }
) )  =  ( M `  { x  e.  O  |  -.  ph } )
2221eqeq1i 2445 . 2  |-  ( ( M `  ( U. dom  M  \  { x  e.  O  |  ph }
) )  =  0  <-> 
( M `  {
x  e.  O  |  -.  ph } )  =  0 )
2317, 22syl6bb 261 1  |-  ( M  e.  U. ran measures  ->  ( { x  e.  O  |  ph }a.e. M  <->  ( M `  { x  e.  O  |  -.  ph } )  =  0 ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   {crab 2714   _Vcvv 2967    \ cdif 3320   U.cuni 4086   class class class wbr 4287   dom cdm 4835   ran crn 4836   ` cfv 5413   0cc0 9274  measurescmeas 26561  a.e.cae 26605
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pr 4526  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-br 4288  df-opab 4346  df-cnv 4843  df-dm 4845  df-rn 4846  df-iota 5376  df-fv 5421  df-ae 26607
This theorem is referenced by:  truae  26611  aean  26612
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