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Theorem brae 30820
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 478 . . . . 5 ((𝑎 = 𝐴𝑚 = 𝑀) → 𝑚 = 𝑀)
21dmeqd 5529 . . . . . . 7 ((𝑎 = 𝐴𝑚 = 𝑀) → dom 𝑚 = dom 𝑀)
32unieqd 4638 . . . . . 6 ((𝑎 = 𝐴𝑚 = 𝑀) → dom 𝑚 = dom 𝑀)
4 simpl 475 . . . . . 6 ((𝑎 = 𝐴𝑚 = 𝑀) → 𝑎 = 𝐴)
53, 4difeq12d 3927 . . . . 5 ((𝑎 = 𝐴𝑚 = 𝑀) → ( dom 𝑚𝑎) = ( dom 𝑀𝐴))
61, 5fveq12d 6418 . . . 4 ((𝑎 = 𝐴𝑚 = 𝑀) → (𝑚‘( dom 𝑚𝑎)) = (𝑀‘( dom 𝑀𝐴)))
76eqeq1d 2801 . . 3 ((𝑎 = 𝐴𝑚 = 𝑀) → ((𝑚‘( dom 𝑚𝑎)) = 0 ↔ (𝑀‘( dom 𝑀𝐴)) = 0))
8 df-ae 30818 . . 3 a.e. = {⟨𝑎, 𝑚⟩ ∣ (𝑚‘( dom 𝑚𝑎)) = 0}
97, 8brabga 5185 . 2 ((𝐴 ∈ dom 𝑀𝑀 ran measures) → (𝐴a.e.𝑀 ↔ (𝑀‘( dom 𝑀𝐴)) = 0))
109ancoms 451 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 198  wa 385   = wceq 1653  wcel 2157  cdif 3766   cuni 4628   class class class wbr 4843  dom cdm 5312  ran crn 5313  cfv 6101  0cc0 10224  measurescmeas 30774  a.e.cae 30816
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-dm 5322  df-iota 6064  df-fv 6109  df-ae 30818
This theorem is referenced by: (None)
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