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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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