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Theorem braew 33240
Description: 'almost everywhere' relation for a measure 𝑀 and a property 𝜑 (Contributed by Thierry Arnoux, 20-Oct-2017.)
Hypothesis
Ref Expression
braew.1 dom 𝑀 = 𝑂
Assertion
Ref Expression
braew (𝑀 ran measures → ({𝑥𝑂𝜑}a.e.𝑀 ↔ (𝑀‘{𝑥𝑂 ∣ ¬ 𝜑}) = 0))
Distinct variable group:   𝑥,𝑂
Allowed substitution hints:   𝜑(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 dom 𝑀 = 𝑂
2 dmexg 7894 . . . . . 6 (𝑀 ran measures → dom 𝑀 ∈ V)
32uniexd 7732 . . . . 5 (𝑀 ran measures → dom 𝑀 ∈ V)
41, 3eqeltrrid 2839 . . . 4 (𝑀 ran measures → 𝑂 ∈ V)
5 rabexg 5332 . . . 4 (𝑂 ∈ V → {𝑥𝑂𝜑} ∈ V)
64, 5syl 17 . . 3 (𝑀 ran measures → {𝑥𝑂𝜑} ∈ V)
7 simpr 486 . . . . . 6 ((𝑎 = {𝑥𝑂𝜑} ∧ 𝑚 = 𝑀) → 𝑚 = 𝑀)
87dmeqd 5906 . . . . . . . 8 ((𝑎 = {𝑥𝑂𝜑} ∧ 𝑚 = 𝑀) → dom 𝑚 = dom 𝑀)
98unieqd 4923 . . . . . . 7 ((𝑎 = {𝑥𝑂𝜑} ∧ 𝑚 = 𝑀) → dom 𝑚 = dom 𝑀)
10 simpl 484 . . . . . . 7 ((𝑎 = {𝑥𝑂𝜑} ∧ 𝑚 = 𝑀) → 𝑎 = {𝑥𝑂𝜑})
119, 10difeq12d 4124 . . . . . 6 ((𝑎 = {𝑥𝑂𝜑} ∧ 𝑚 = 𝑀) → ( dom 𝑚𝑎) = ( dom 𝑀 ∖ {𝑥𝑂𝜑}))
127, 11fveq12d 6899 . . . . 5 ((𝑎 = {𝑥𝑂𝜑} ∧ 𝑚 = 𝑀) → (𝑚‘( dom 𝑚𝑎)) = (𝑀‘( dom 𝑀 ∖ {𝑥𝑂𝜑})))
1312eqeq1d 2735 . . . 4 ((𝑎 = {𝑥𝑂𝜑} ∧ 𝑚 = 𝑀) → ((𝑚‘( dom 𝑚𝑎)) = 0 ↔ (𝑀‘( dom 𝑀 ∖ {𝑥𝑂𝜑})) = 0))
14 df-ae 33237 . . . 4 a.e. = {⟨𝑎, 𝑚⟩ ∣ (𝑚‘( dom 𝑚𝑎)) = 0}
1513, 14brabga 5535 . . 3 (({𝑥𝑂𝜑} ∈ V ∧ 𝑀 ran measures) → ({𝑥𝑂𝜑}a.e.𝑀 ↔ (𝑀‘( dom 𝑀 ∖ {𝑥𝑂𝜑})) = 0))
166, 15mpancom 687 . 2 (𝑀 ran measures → ({𝑥𝑂𝜑}a.e.𝑀 ↔ (𝑀‘( dom 𝑀 ∖ {𝑥𝑂𝜑})) = 0))
171difeq1i 4119 . . . . 5 ( dom 𝑀 ∖ {𝑥𝑂𝜑}) = (𝑂 ∖ {𝑥𝑂𝜑})
18 notrab 4312 . . . . 5 (𝑂 ∖ {𝑥𝑂𝜑}) = {𝑥𝑂 ∣ ¬ 𝜑}
1917, 18eqtri 2761 . . . 4 ( dom 𝑀 ∖ {𝑥𝑂𝜑}) = {𝑥𝑂 ∣ ¬ 𝜑}
2019fveq2i 6895 . . 3 (𝑀‘( dom 𝑀 ∖ {𝑥𝑂𝜑})) = (𝑀‘{𝑥𝑂 ∣ ¬ 𝜑})
2120eqeq1i 2738 . 2 ((𝑀‘( dom 𝑀 ∖ {𝑥𝑂𝜑})) = 0 ↔ (𝑀‘{𝑥𝑂 ∣ ¬ 𝜑}) = 0)
2216, 21bitrdi 287 1 (𝑀 ran measures → ({𝑥𝑂𝜑}a.e.𝑀 ↔ (𝑀‘{𝑥𝑂 ∣ ¬ 𝜑}) = 0))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  {crab 3433  Vcvv 3475  cdif 3946   cuni 4909   class class class wbr 5149  dom cdm 5677  ran crn 5678  cfv 6544  0cc0 11110  measurescmeas 33193  a.e.cae 33235
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-cnv 5685  df-dm 5687  df-rn 5688  df-iota 6496  df-fv 6552  df-ae 33237
This theorem is referenced by:  truae  33241  aean  33242
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