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Theorem braew 29017
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 6682 . . . . . 6  |-  ( M  e.  U. ran measures  ->  dom  M  e.  _V )
3 uniexg 6546 . . . . . 6  |-  ( dom 
M  e.  _V  ->  U.
dom  M  e.  _V )
42, 3syl 17 . . . . 5  |-  ( M  e.  U. ran measures  ->  U. dom  M  e.  _V )
51, 4syl5eqelr 2511 . . . 4  |-  ( M  e.  U. ran measures  ->  O  e.  _V )
6 rabexg 4517 . . . 4  |-  ( O  e.  _V  ->  { x  e.  O  |  ph }  e.  _V )
75, 6syl 17 . . 3  |-  ( M  e.  U. ran measures  ->  { x  e.  O  |  ph }  e.  _V )
8 simpr 462 . . . . . 6  |-  ( ( a  =  { x  e.  O  |  ph }  /\  m  =  M
)  ->  m  =  M )
98dmeqd 4999 . . . . . . . 8  |-  ( ( a  =  { x  e.  O  |  ph }  /\  m  =  M
)  ->  dom  m  =  dom  M )
109unieqd 4172 . . . . . . 7  |-  ( ( a  =  { x  e.  O  |  ph }  /\  m  =  M
)  ->  U. dom  m  =  U. dom  M )
11 simpl 458 . . . . . . 7  |-  ( ( a  =  { x  e.  O  |  ph }  /\  m  =  M
)  ->  a  =  { x  e.  O  |  ph } )
1210, 11difeq12d 3527 . . . . . 6  |-  ( ( a  =  { x  e.  O  |  ph }  /\  m  =  M
)  ->  ( U. dom  m  \  a )  =  ( U. dom  M 
\  { x  e.  O  |  ph }
) )
138, 12fveq12d 5831 . . . . 5  |-  ( ( a  =  { x  e.  O  |  ph }  /\  m  =  M
)  ->  ( m `  ( U. dom  m  \  a ) )  =  ( M `  ( U. dom  M  \  {
x  e.  O  |  ph } ) ) )
1413eqeq1d 2430 . . . 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 29014 . . . 4  |- a.e.  =  { <. a ,  m >.  |  ( m `  ( U. dom  m  \  a
) )  =  0 }
1614, 15brabga 4677 . . 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 673 . 2  |-  ( M  e.  U. ran measures  ->  ( { x  e.  O  |  ph }a.e. M  <->  ( M `  ( U. dom  M  \  { x  e.  O  |  ph } ) )  =  0 ) )
181difeq1i 3522 . . . . 5  |-  ( U. dom  M  \  { x  e.  O  |  ph }
)  =  ( O 
\  { x  e.  O  |  ph }
)
19 notrab 3693 . . . . 5  |-  ( O 
\  { x  e.  O  |  ph }
)  =  { x  e.  O  |  -.  ph }
2018, 19eqtri 2450 . . . 4  |-  ( U. dom  M  \  { x  e.  O  |  ph }
)  =  { x  e.  O  |  -.  ph }
2120fveq2i 5828 . . 3  |-  ( M `
 ( U. dom  M 
\  { x  e.  O  |  ph }
) )  =  ( M `  { x  e.  O  |  -.  ph } )
2221eqeq1i 2433 . 2  |-  ( ( M `  ( U. dom  M  \  { x  e.  O  |  ph }
) )  =  0  <-> 
( M `  {
x  e.  O  |  -.  ph } )  =  0 )
2317, 22syl6bb 264 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 187    /\ wa 370    = wceq 1437    e. wcel 1872   {crab 2718   _Vcvv 3022    \ 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-8 1874  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  ax-un 6541
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-cnv 4804  df-dm 4806  df-rn 4807  df-iota 5508  df-fv 5552  df-ae 29014
This theorem is referenced by:  truae  29018  aean  29019
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