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Theorem brae 30432
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 476 . . . . 5 ((𝑎 = 𝐴𝑚 = 𝑀) → 𝑚 = 𝑀)
21dmeqd 5358 . . . . . . 7 ((𝑎 = 𝐴𝑚 = 𝑀) → dom 𝑚 = dom 𝑀)
32unieqd 4478 . . . . . 6 ((𝑎 = 𝐴𝑚 = 𝑀) → dom 𝑚 = dom 𝑀)
4 simpl 472 . . . . . 6 ((𝑎 = 𝐴𝑚 = 𝑀) → 𝑎 = 𝐴)
53, 4difeq12d 3762 . . . . 5 ((𝑎 = 𝐴𝑚 = 𝑀) → ( dom 𝑚𝑎) = ( dom 𝑀𝐴))
61, 5fveq12d 6235 . . . 4 ((𝑎 = 𝐴𝑚 = 𝑀) → (𝑚‘( dom 𝑚𝑎)) = (𝑀‘( dom 𝑀𝐴)))
76eqeq1d 2653 . . 3 ((𝑎 = 𝐴𝑚 = 𝑀) → ((𝑚‘( dom 𝑚𝑎)) = 0 ↔ (𝑀‘( dom 𝑀𝐴)) = 0))
8 df-ae 30430 . . 3 a.e. = {⟨𝑎, 𝑚⟩ ∣ (𝑚‘( dom 𝑚𝑎)) = 0}
97, 8brabga 5018 . 2 ((𝐴 ∈ dom 𝑀𝑀 ran measures) → (𝐴a.e.𝑀 ↔ (𝑀‘( dom 𝑀𝐴)) = 0))
109ancoms 468 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 196  wa 383   = wceq 1523  wcel 2030  cdif 3604   cuni 4468   class class class wbr 4685  dom cdm 5143  ran crn 5144  cfv 5926  0cc0 9974  measurescmeas 30386  a.e.cae 30428
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-dm 5153  df-iota 5889  df-fv 5934  df-ae 30430
This theorem is referenced by: (None)
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