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Theorem brae 31500
Description: 'almost everywhere' relation for a measure and a measurable set 𝐴. (Contributed by Thierry Arnoux, 20-Oct-2017.)
Assertion
Ref Expression
brae ((𝑀 ran measures ∧ 𝐴 ∈ dom 𝑀) → (𝐴a.e.𝑀 ↔ (𝑀‘( dom 𝑀𝐴)) = 0))

Proof of Theorem brae
Dummy variables 𝑚 𝑎 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 487 . . . . 5 ((𝑎 = 𝐴𝑚 = 𝑀) → 𝑚 = 𝑀)
21dmeqd 5773 . . . . . . 7 ((𝑎 = 𝐴𝑚 = 𝑀) → dom 𝑚 = dom 𝑀)
32unieqd 4851 . . . . . 6 ((𝑎 = 𝐴𝑚 = 𝑀) → dom 𝑚 = dom 𝑀)
4 simpl 485 . . . . . 6 ((𝑎 = 𝐴𝑚 = 𝑀) → 𝑎 = 𝐴)
53, 4difeq12d 4099 . . . . 5 ((𝑎 = 𝐴𝑚 = 𝑀) → ( dom 𝑚𝑎) = ( dom 𝑀𝐴))
61, 5fveq12d 6676 . . . 4 ((𝑎 = 𝐴𝑚 = 𝑀) → (𝑚‘( dom 𝑚𝑎)) = (𝑀‘( dom 𝑀𝐴)))
76eqeq1d 2823 . . 3 ((𝑎 = 𝐴𝑚 = 𝑀) → ((𝑚‘( dom 𝑚𝑎)) = 0 ↔ (𝑀‘( dom 𝑀𝐴)) = 0))
8 df-ae 31498 . . 3 a.e. = {⟨𝑎, 𝑚⟩ ∣ (𝑚‘( dom 𝑚𝑎)) = 0}
97, 8brabga 5420 . 2 ((𝐴 ∈ dom 𝑀𝑀 ran measures) → (𝐴a.e.𝑀 ↔ (𝑀‘( dom 𝑀𝐴)) = 0))
109ancoms 461 1 ((𝑀 ran measures ∧ 𝐴 ∈ dom 𝑀) → (𝐴a.e.𝑀 ↔ (𝑀‘( dom 𝑀𝐴)) = 0))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398   = wceq 1533  wcel 2110  cdif 3932   cuni 4837   class class class wbr 5065  dom cdm 5554  ran crn 5555  cfv 6354  0cc0 10536  measurescmeas 31454  a.e.cae 31496
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pr 5329
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-br 5066  df-opab 5128  df-dm 5564  df-iota 6313  df-fv 6362  df-ae 31498
This theorem is referenced by: (None)
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