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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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