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Theorem truae 30646
 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 148 . . . . . . 7 (𝜑 → (¬ 𝜓𝑥 ∈ ∅))
32ralrimivw 3116 . . . . . 6 (𝜑 → ∀𝑥𝑂𝜓𝑥 ∈ ∅))
4 rabss 3828 . . . . . 6 ({𝑥𝑂 ∣ ¬ 𝜓} ⊆ ∅ ↔ ∀𝑥𝑂𝜓𝑥 ∈ ∅))
53, 4sylibr 224 . . . . 5 (𝜑 → {𝑥𝑂 ∣ ¬ 𝜓} ⊆ ∅)
6 ss0 4118 . . . . 5 ({𝑥𝑂 ∣ ¬ 𝜓} ⊆ ∅ → {𝑥𝑂 ∣ ¬ 𝜓} = ∅)
75, 6syl 17 . . . 4 (𝜑 → {𝑥𝑂 ∣ ¬ 𝜓} = ∅)
87fveq2d 6336 . . 3 (𝜑 → (𝑀‘{𝑥𝑂 ∣ ¬ 𝜓}) = (𝑀‘∅))
9 truae.2 . . . 4 (𝜑𝑀 ran measures)
10 measbasedom 30605 . . . . 5 (𝑀 ran measures ↔ 𝑀 ∈ (measures‘dom 𝑀))
11 measvnul 30609 . . . . 5 (𝑀 ∈ (measures‘dom 𝑀) → (𝑀‘∅) = 0)
1210, 11sylbi 207 . . . 4 (𝑀 ran measures → (𝑀‘∅) = 0)
139, 12syl 17 . . 3 (𝜑 → (𝑀‘∅) = 0)
148, 13eqtrd 2805 . 2 (𝜑 → (𝑀‘{𝑥𝑂 ∣ ¬ 𝜓}) = 0)
15 truae.1 . . . 4 dom 𝑀 = 𝑂
1615braew 30645 . . 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 247 1 (𝜑 → {𝑥𝑂𝜓}a.e.𝑀)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   = wceq 1631   ∈ wcel 2145  ∀wral 3061  {crab 3065   ⊆ wss 3723  ∅c0 4063  ∪ cuni 4574   class class class wbr 4786  dom cdm 5249  ran crn 5250  ‘cfv 6031  0cc0 10138  measurescmeas 30598  a.e.cae 30640 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fv 6039  df-ov 6796  df-esum 30430  df-meas 30599  df-ae 30642 This theorem is referenced by: (None)
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