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Theorem braew 30614
 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 7262 . . . . . 6 (𝑀 ran measures → dom 𝑀 ∈ V)
3 uniexg 7120 . . . . . 6 (dom 𝑀 ∈ V → dom 𝑀 ∈ V)
42, 3syl 17 . . . . 5 (𝑀 ran measures → dom 𝑀 ∈ V)
51, 4syl5eqelr 2844 . . . 4 (𝑀 ran measures → 𝑂 ∈ V)
6 rabexg 4963 . . . 4 (𝑂 ∈ V → {𝑥𝑂𝜑} ∈ V)
75, 6syl 17 . . 3 (𝑀 ran measures → {𝑥𝑂𝜑} ∈ V)
8 simpr 479 . . . . . 6 ((𝑎 = {𝑥𝑂𝜑} ∧ 𝑚 = 𝑀) → 𝑚 = 𝑀)
98dmeqd 5481 . . . . . . . 8 ((𝑎 = {𝑥𝑂𝜑} ∧ 𝑚 = 𝑀) → dom 𝑚 = dom 𝑀)
109unieqd 4598 . . . . . . 7 ((𝑎 = {𝑥𝑂𝜑} ∧ 𝑚 = 𝑀) → dom 𝑚 = dom 𝑀)
11 simpl 474 . . . . . . 7 ((𝑎 = {𝑥𝑂𝜑} ∧ 𝑚 = 𝑀) → 𝑎 = {𝑥𝑂𝜑})
1210, 11difeq12d 3872 . . . . . 6 ((𝑎 = {𝑥𝑂𝜑} ∧ 𝑚 = 𝑀) → ( dom 𝑚𝑎) = ( dom 𝑀 ∖ {𝑥𝑂𝜑}))
138, 12fveq12d 6358 . . . . 5 ((𝑎 = {𝑥𝑂𝜑} ∧ 𝑚 = 𝑀) → (𝑚‘( dom 𝑚𝑎)) = (𝑀‘( dom 𝑀 ∖ {𝑥𝑂𝜑})))
1413eqeq1d 2762 . . . 4 ((𝑎 = {𝑥𝑂𝜑} ∧ 𝑚 = 𝑀) → ((𝑚‘( dom 𝑚𝑎)) = 0 ↔ (𝑀‘( dom 𝑀 ∖ {𝑥𝑂𝜑})) = 0))
15 df-ae 30611 . . . 4 a.e. = {⟨𝑎, 𝑚⟩ ∣ (𝑚‘( dom 𝑚𝑎)) = 0}
1614, 15brabga 5139 . . 3 (({𝑥𝑂𝜑} ∈ V ∧ 𝑀 ran measures) → ({𝑥𝑂𝜑}a.e.𝑀 ↔ (𝑀‘( dom 𝑀 ∖ {𝑥𝑂𝜑})) = 0))
177, 16mpancom 706 . 2 (𝑀 ran measures → ({𝑥𝑂𝜑}a.e.𝑀 ↔ (𝑀‘( dom 𝑀 ∖ {𝑥𝑂𝜑})) = 0))
181difeq1i 3867 . . . . 5 ( dom 𝑀 ∖ {𝑥𝑂𝜑}) = (𝑂 ∖ {𝑥𝑂𝜑})
19 notrab 4047 . . . . 5 (𝑂 ∖ {𝑥𝑂𝜑}) = {𝑥𝑂 ∣ ¬ 𝜑}
2018, 19eqtri 2782 . . . 4 ( dom 𝑀 ∖ {𝑥𝑂𝜑}) = {𝑥𝑂 ∣ ¬ 𝜑}
2120fveq2i 6355 . . 3 (𝑀‘( dom 𝑀 ∖ {𝑥𝑂𝜑})) = (𝑀‘{𝑥𝑂 ∣ ¬ 𝜑})
2221eqeq1i 2765 . 2 ((𝑀‘( dom 𝑀 ∖ {𝑥𝑂𝜑})) = 0 ↔ (𝑀‘{𝑥𝑂 ∣ ¬ 𝜑}) = 0)
2317, 22syl6bb 276 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 196   ∧ wa 383   = wceq 1632   ∈ wcel 2139  {crab 3054  Vcvv 3340   ∖ cdif 3712  ∪ cuni 4588   class class class wbr 4804  dom cdm 5266  ran crn 5267  ‘cfv 6049  0cc0 10128  measurescmeas 30567  a.e.cae 30609 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055  ax-un 7114 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-cnv 5274  df-dm 5276  df-rn 5277  df-iota 6012  df-fv 6057  df-ae 30611 This theorem is referenced by:  truae  30615  aean  30616
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