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Theorem braew 31496
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 7607 . . . . . 6 (𝑀 ran measures → dom 𝑀 ∈ V)
32uniexd 7462 . . . . 5 (𝑀 ran measures → dom 𝑀 ∈ V)
41, 3eqeltrrid 2918 . . . 4 (𝑀 ran measures → 𝑂 ∈ V)
5 rabexg 5226 . . . 4 (𝑂 ∈ V → {𝑥𝑂𝜑} ∈ V)
64, 5syl 17 . . 3 (𝑀 ran measures → {𝑥𝑂𝜑} ∈ V)
7 simpr 487 . . . . . 6 ((𝑎 = {𝑥𝑂𝜑} ∧ 𝑚 = 𝑀) → 𝑚 = 𝑀)
87dmeqd 5768 . . . . . . . 8 ((𝑎 = {𝑥𝑂𝜑} ∧ 𝑚 = 𝑀) → dom 𝑚 = dom 𝑀)
98unieqd 4841 . . . . . . 7 ((𝑎 = {𝑥𝑂𝜑} ∧ 𝑚 = 𝑀) → dom 𝑚 = dom 𝑀)
10 simpl 485 . . . . . . 7 ((𝑎 = {𝑥𝑂𝜑} ∧ 𝑚 = 𝑀) → 𝑎 = {𝑥𝑂𝜑})
119, 10difeq12d 4099 . . . . . 6 ((𝑎 = {𝑥𝑂𝜑} ∧ 𝑚 = 𝑀) → ( dom 𝑚𝑎) = ( dom 𝑀 ∖ {𝑥𝑂𝜑}))
127, 11fveq12d 6671 . . . . 5 ((𝑎 = {𝑥𝑂𝜑} ∧ 𝑚 = 𝑀) → (𝑚‘( dom 𝑚𝑎)) = (𝑀‘( dom 𝑀 ∖ {𝑥𝑂𝜑})))
1312eqeq1d 2823 . . . 4 ((𝑎 = {𝑥𝑂𝜑} ∧ 𝑚 = 𝑀) → ((𝑚‘( dom 𝑚𝑎)) = 0 ↔ (𝑀‘( dom 𝑀 ∖ {𝑥𝑂𝜑})) = 0))
14 df-ae 31493 . . . 4 a.e. = {⟨𝑎, 𝑚⟩ ∣ (𝑚‘( dom 𝑚𝑎)) = 0}
1513, 14brabga 5413 . . 3 (({𝑥𝑂𝜑} ∈ V ∧ 𝑀 ran measures) → ({𝑥𝑂𝜑}a.e.𝑀 ↔ (𝑀‘( dom 𝑀 ∖ {𝑥𝑂𝜑})) = 0))
166, 15mpancom 686 . 2 (𝑀 ran measures → ({𝑥𝑂𝜑}a.e.𝑀 ↔ (𝑀‘( dom 𝑀 ∖ {𝑥𝑂𝜑})) = 0))
171difeq1i 4094 . . . . 5 ( dom 𝑀 ∖ {𝑥𝑂𝜑}) = (𝑂 ∖ {𝑥𝑂𝜑})
18 notrab 4279 . . . . 5 (𝑂 ∖ {𝑥𝑂𝜑}) = {𝑥𝑂 ∣ ¬ 𝜑}
1917, 18eqtri 2844 . . . 4 ( dom 𝑀 ∖ {𝑥𝑂𝜑}) = {𝑥𝑂 ∣ ¬ 𝜑}
2019fveq2i 6667 . . 3 (𝑀‘( dom 𝑀 ∖ {𝑥𝑂𝜑})) = (𝑀‘{𝑥𝑂 ∣ ¬ 𝜑})
2120eqeq1i 2826 . 2 ((𝑀‘( dom 𝑀 ∖ {𝑥𝑂𝜑})) = 0 ↔ (𝑀‘{𝑥𝑂 ∣ ¬ 𝜑}) = 0)
2216, 21syl6bb 289 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 208  wa 398   = wceq 1533  wcel 2110  {crab 3142  Vcvv 3494  cdif 3932   cuni 4831   class class class wbr 5058  dom cdm 5549  ran crn 5550  cfv 6349  0cc0 10531  measurescmeas 31449  a.e.cae 31491
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 5195  ax-nul 5202  ax-pr 5321  ax-un 7455
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-rex 3144  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 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-opab 5121  df-cnv 5557  df-dm 5559  df-rn 5560  df-iota 6308  df-fv 6357  df-ae 31493
This theorem is referenced by:  truae  31497  aean  31498
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