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Theorem truae 31576
Description: A truth holds almost everywhere. (Contributed by Thierry Arnoux, 20-Oct-2017.)
Hypotheses
Ref Expression
truae.1 dom 𝑀 = 𝑂
truae.2 (𝜑𝑀 ran measures)
truae.3 (𝜑𝜓)
Assertion
Ref Expression
truae (𝜑 → {𝑥𝑂𝜓}a.e.𝑀)
Distinct variable groups:   𝑥,𝑂   𝜑,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝑀(𝑥)

Proof of Theorem truae
StepHypRef Expression
1 truae.3 . . . . . . . 8 (𝜑𝜓)
21pm2.24d 154 . . . . . . 7 (𝜑 → (¬ 𝜓𝑥 ∈ ∅))
32ralrimivw 3175 . . . . . 6 (𝜑 → ∀𝑥𝑂𝜓𝑥 ∈ ∅))
4 rabss 4023 . . . . . 6 ({𝑥𝑂 ∣ ¬ 𝜓} ⊆ ∅ ↔ ∀𝑥𝑂𝜓𝑥 ∈ ∅))
53, 4sylibr 237 . . . . 5 (𝜑 → {𝑥𝑂 ∣ ¬ 𝜓} ⊆ ∅)
6 ss0 4324 . . . . 5 ({𝑥𝑂 ∣ ¬ 𝜓} ⊆ ∅ → {𝑥𝑂 ∣ ¬ 𝜓} = ∅)
75, 6syl 17 . . . 4 (𝜑 → {𝑥𝑂 ∣ ¬ 𝜓} = ∅)
87fveq2d 6656 . . 3 (𝜑 → (𝑀‘{𝑥𝑂 ∣ ¬ 𝜓}) = (𝑀‘∅))
9 truae.2 . . . 4 (𝜑𝑀 ran measures)
10 measbasedom 31535 . . . . 5 (𝑀 ran measures ↔ 𝑀 ∈ (measures‘dom 𝑀))
11 measvnul 31539 . . . . 5 (𝑀 ∈ (measures‘dom 𝑀) → (𝑀‘∅) = 0)
1210, 11sylbi 220 . . . 4 (𝑀 ran measures → (𝑀‘∅) = 0)
139, 12syl 17 . . 3 (𝜑 → (𝑀‘∅) = 0)
148, 13eqtrd 2857 . 2 (𝜑 → (𝑀‘{𝑥𝑂 ∣ ¬ 𝜓}) = 0)
15 truae.1 . . . 4 dom 𝑀 = 𝑂
1615braew 31575 . . 3 (𝑀 ran measures → ({𝑥𝑂𝜓}a.e.𝑀 ↔ (𝑀‘{𝑥𝑂 ∣ ¬ 𝜓}) = 0))
179, 16syl 17 . 2 (𝜑 → ({𝑥𝑂𝜓}a.e.𝑀 ↔ (𝑀‘{𝑥𝑂 ∣ ¬ 𝜓}) = 0))
1814, 17mpbird 260 1 (𝜑 → {𝑥𝑂𝜓}a.e.𝑀)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209   = wceq 1538  wcel 2114  wral 3130  {crab 3134  wss 3908  c0 4265   cuni 4813   class class class wbr 5042  dom cdm 5532  ran crn 5533  cfv 6334  0cc0 10526  measurescmeas 31528  a.e.cae 31570
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-fv 6342  df-ov 7143  df-esum 31361  df-meas 31529  df-ae 31572
This theorem is referenced by: (None)
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