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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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