Theorem truae 31612

Description: A truth holds almost everywhere. (Contributed by Thierry Arnoux, 20-Oct-2017.)

Hypotheses

Ref	Expression
truae.1	⊢ ∪ dom 𝑀 = 𝑂
truae.2	⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)
truae.3	⊢ (𝜑 → 𝜓)

Assertion

Ref	Expression
truae	⊢ (𝜑 → {𝑥 ∈ 𝑂 ∣ 𝜓}a.e.𝑀)

Distinct variable groups:   𝑥,𝑂   𝜑,𝑥

Allowed substitution hints:   𝜓(𝑥)   𝑀(𝑥)

Proof of Theorem truae

Step	Hyp	Ref	Expression
1		truae.3	. . . . . . . 8 ⊢ (𝜑 → 𝜓)
2	1	pm2.24d 154	. . . . . . 7 ⊢ (𝜑 → (¬ 𝜓 → 𝑥 ∈ ∅))
3	2	ralrimivw 3150	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝑂 (¬ 𝜓 → 𝑥 ∈ ∅))
4		rabss 3999	. . . . . 6 ⊢ ({𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ⊆ ∅ ↔ ∀𝑥 ∈ 𝑂 (¬ 𝜓 → 𝑥 ∈ ∅))
5	3, 4	sylibr 237	. . . . 5 ⊢ (𝜑 → {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ⊆ ∅)
6		ss0 4306	. . . . 5 ⊢ ({𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ⊆ ∅ → {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} = ∅)
7	5, 6	syl 17	. . . 4 ⊢ (𝜑 → {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} = ∅)
8	7	fveq2d 6649	. . 3 ⊢ (𝜑 → (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜓}) = (𝑀‘∅))
9		truae.2	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)
10		measbasedom 31571	. . . . 5 ⊢ (𝑀 ∈ ∪ ran measures ↔ 𝑀 ∈ (measures‘dom 𝑀))
11		measvnul 31575	. . . . 5 ⊢ (𝑀 ∈ (measures‘dom 𝑀) → (𝑀‘∅) = 0)
12	10, 11	sylbi 220	. . . 4 ⊢ (𝑀 ∈ ∪ ran measures → (𝑀‘∅) = 0)
13	9, 12	syl 17	. . 3 ⊢ (𝜑 → (𝑀‘∅) = 0)
14	8, 13	eqtrd 2833	. 2 ⊢ (𝜑 → (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜓}) = 0)
15		truae.1	. . . 4 ⊢ ∪ dom 𝑀 = 𝑂
16	15	braew 31611	. . 3 ⊢ (𝑀 ∈ ∪ ran measures → ({𝑥 ∈ 𝑂 ∣ 𝜓}a.e.𝑀 ↔ (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜓}) = 0))
17	9, 16	syl 17	. 2 ⊢ (𝜑 → ({𝑥 ∈ 𝑂 ∣ 𝜓}a.e.𝑀 ↔ (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜓}) = 0))
18	14, 17	mpbird 260	1 ⊢ (𝜑 → {𝑥 ∈ 𝑂 ∣ 𝜓}a.e.𝑀)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   = wceq 1538   ∈ wcel 2111  ∀wral 3106  {crab 3110   ⊆ wss 3881  ∅c0 4243  ∪ cuni 4800   class class class wbr 5030  dom cdm 5519  ran crn 5520  ‘cfv 6324  0cc0 10526  measurescmeas 31564  a.e.cae 31606

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 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

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 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-fv 6332  df-ov 7138  df-esum 31397  df-meas 31565  df-ae 31608

This theorem is referenced by: (None)