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Theorem braew 28039
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 6726 . . . . . 6  |-  ( M  e.  U. ran measures  ->  dom  M  e.  _V )
3 uniexg 6592 . . . . . 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 2560 . . . 4  |-  ( M  e.  U. ran measures  ->  O  e.  _V )
6 rabexg 4603 . . . 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 5211 . . . . . . . 8  |-  ( ( a  =  { x  e.  O  |  ph }  /\  m  =  M
)  ->  dom  m  =  dom  M )
109unieqd 4261 . . . . . . 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 3628 . . . . . 6  |-  ( ( a  =  { x  e.  O  |  ph }  /\  m  =  M
)  ->  ( U. dom  m  \  a )  =  ( U. dom  M 
\  { x  e.  O  |  ph }
) )
138, 12fveq12d 5878 . . . . 5  |-  ( ( a  =  { x  e.  O  |  ph }  /\  m  =  M
)  ->  ( m `  ( U. dom  m  \  a ) )  =  ( M `  ( U. dom  M  \  {
x  e.  O  |  ph } ) ) )
1413eqeq1d 2469 . . . 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 28036 . . . 4  |- a.e.  =  { <. a ,  m >.  |  ( m `  ( U. dom  m  \  a
) )  =  0 }
1614, 15brabga 4767 . . 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 3623 . . . . 5  |-  ( U. dom  M  \  { x  e.  O  |  ph }
)  =  ( O 
\  { x  e.  O  |  ph }
)
19 notrab 3780 . . . . 5  |-  ( O 
\  { x  e.  O  |  ph }
)  =  { x  e.  O  |  -.  ph }
2018, 19eqtri 2496 . . . 4  |-  ( U. dom  M  \  { x  e.  O  |  ph }
)  =  { x  e.  O  |  -.  ph }
2120fveq2i 5875 . . 3  |-  ( M `
 ( U. dom  M 
\  { x  e.  O  |  ph }
) )  =  ( M `  { x  e.  O  |  -.  ph } )
2221eqeq1i 2474 . 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 1379    e. wcel 1767   {crab 2821   _Vcvv 3118    \ cdif 3478   U.cuni 4251   class class class wbr 4453   dom cdm 5005   ran crn 5006   ` cfv 5594   0cc0 9504  measurescmeas 27991  a.e.cae 28034
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-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692  ax-un 6587
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 2822  df-rex 2823  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-cnv 5013  df-dm 5015  df-rn 5016  df-iota 5557  df-fv 5602  df-ae 28036
This theorem is referenced by:  truae  28040  aean  28041
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