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Theorem braew 26822
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 6622 . . . . . 6  |-  ( M  e.  U. ran measures  ->  dom  M  e.  _V )
3 uniexg 6490 . . . . . 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 2547 . . . 4  |-  ( M  e.  U. ran measures  ->  O  e.  _V )
6 rabexg 4553 . . . 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 5153 . . . . . . . 8  |-  ( ( a  =  { x  e.  O  |  ph }  /\  m  =  M
)  ->  dom  m  =  dom  M )
109unieqd 4212 . . . . . . 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 3586 . . . . . 6  |-  ( ( a  =  { x  e.  O  |  ph }  /\  m  =  M
)  ->  ( U. dom  m  \  a )  =  ( U. dom  M 
\  { x  e.  O  |  ph }
) )
138, 12fveq12d 5808 . . . . 5  |-  ( ( a  =  { x  e.  O  |  ph }  /\  m  =  M
)  ->  ( m `  ( U. dom  m  \  a ) )  =  ( M `  ( U. dom  M  \  {
x  e.  O  |  ph } ) ) )
1413eqeq1d 2456 . . . 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 26819 . . . 4  |- a.e.  =  { <. a ,  m >.  |  ( m `  ( U. dom  m  \  a
) )  =  0 }
1614, 15brabga 4714 . . 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 3581 . . . . 5  |-  ( U. dom  M  \  { x  e.  O  |  ph }
)  =  ( O 
\  { x  e.  O  |  ph }
)
19 notrab 3738 . . . . 5  |-  ( O 
\  { x  e.  O  |  ph }
)  =  { x  e.  O  |  -.  ph }
2018, 19eqtri 2483 . . . 4  |-  ( U. dom  M  \  { x  e.  O  |  ph }
)  =  { x  e.  O  |  -.  ph }
2120fveq2i 5805 . . 3  |-  ( M `
 ( U. dom  M 
\  { x  e.  O  |  ph }
) )  =  ( M `  { x  e.  O  |  -.  ph } )
2221eqeq1i 2461 . 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 1370    e. wcel 1758   {crab 2803   _Vcvv 3078    \ cdif 3436   U.cuni 4202   class class class wbr 4403   dom cdm 4951   ran crn 4952   ` cfv 5529   0cc0 9396  measurescmeas 26774  a.e.cae 26817
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 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-cnv 4959  df-dm 4961  df-rn 4962  df-iota 5492  df-fv 5537  df-ae 26819
This theorem is referenced by:  truae  26823  aean  26824
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