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Theorem truae 30822
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 149 . . . . . . 7 (𝜑 → (¬ 𝜓𝑥 ∈ ∅))
32ralrimivw 3148 . . . . . 6 (𝜑 → ∀𝑥𝑂𝜓𝑥 ∈ ∅))
4 rabss 3875 . . . . . 6 ({𝑥𝑂 ∣ ¬ 𝜓} ⊆ ∅ ↔ ∀𝑥𝑂𝜓𝑥 ∈ ∅))
53, 4sylibr 226 . . . . 5 (𝜑 → {𝑥𝑂 ∣ ¬ 𝜓} ⊆ ∅)
6 ss0 4170 . . . . 5 ({𝑥𝑂 ∣ ¬ 𝜓} ⊆ ∅ → {𝑥𝑂 ∣ ¬ 𝜓} = ∅)
75, 6syl 17 . . . 4 (𝜑 → {𝑥𝑂 ∣ ¬ 𝜓} = ∅)
87fveq2d 6415 . . 3 (𝜑 → (𝑀‘{𝑥𝑂 ∣ ¬ 𝜓}) = (𝑀‘∅))
9 truae.2 . . . 4 (𝜑𝑀 ran measures)
10 measbasedom 30781 . . . . 5 (𝑀 ran measures ↔ 𝑀 ∈ (measures‘dom 𝑀))
11 measvnul 30785 . . . . 5 (𝑀 ∈ (measures‘dom 𝑀) → (𝑀‘∅) = 0)
1210, 11sylbi 209 . . . 4 (𝑀 ran measures → (𝑀‘∅) = 0)
139, 12syl 17 . . 3 (𝜑 → (𝑀‘∅) = 0)
148, 13eqtrd 2833 . 2 (𝜑 → (𝑀‘{𝑥𝑂 ∣ ¬ 𝜓}) = 0)
15 truae.1 . . . 4 dom 𝑀 = 𝑂
1615braew 30821 . . 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 249 1 (𝜑 → {𝑥𝑂𝜓}a.e.𝑀)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198   = wceq 1653  wcel 2157  wral 3089  {crab 3093  wss 3769  c0 4115   cuni 4628   class class class wbr 4843  dom cdm 5312  ran crn 5313  cfv 6101  0cc0 10224  measurescmeas 30774  a.e.cae 30816
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-fal 1667  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-fv 6109  df-ov 6881  df-esum 30606  df-meas 30775  df-ae 30818
This theorem is referenced by: (None)
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