Theorem truae 34239

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 151	. . . . . . 7 ⊢ (𝜑 → (¬ 𝜓 → 𝑥 ∈ ∅))
3	2	ralrimivw 3130	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝑂 (¬ 𝜓 → 𝑥 ∈ ∅))
4		rabss 4037	. . . . . 6 ⊢ ({𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ⊆ ∅ ↔ ∀𝑥 ∈ 𝑂 (¬ 𝜓 → 𝑥 ∈ ∅))
5	3, 4	sylibr 234	. . . . 5 ⊢ (𝜑 → {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ⊆ ∅)
6		ss0 4367	. . . . 5 ⊢ ({𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ⊆ ∅ → {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} = ∅)
7	5, 6	syl 17	. . . 4 ⊢ (𝜑 → {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} = ∅)
8	7	fveq2d 6864	. . 3 ⊢ (𝜑 → (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜓}) = (𝑀‘∅))
9		truae.2	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)
10		measbasedom 34198	. . . . 5 ⊢ (𝑀 ∈ ∪ ran measures ↔ 𝑀 ∈ (measures‘dom 𝑀))
11		measvnul 34202	. . . . 5 ⊢ (𝑀 ∈ (measures‘dom 𝑀) → (𝑀‘∅) = 0)
12	10, 11	sylbi 217	. . . 4 ⊢ (𝑀 ∈ ∪ ran measures → (𝑀‘∅) = 0)
13	9, 12	syl 17	. . 3 ⊢ (𝜑 → (𝑀‘∅) = 0)
14	8, 13	eqtrd 2765	. 2 ⊢ (𝜑 → (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜓}) = 0)
15		truae.1	. . . 4 ⊢ ∪ dom 𝑀 = 𝑂
16	15	braew 34238	. . 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 257	1 ⊢ (𝜑 → {𝑥 ∈ 𝑂 ∣ 𝜓}a.e.𝑀)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   = wceq 1540   ∈ wcel 2109  ∀wral 3045  {crab 3408   ⊆ wss 3916  ∅c0 4298  ∪ cuni 4873   class class class wbr 5109  dom cdm 5640  ran crn 5641  ‘cfv 6513  0cc0 11074  measurescmeas 34191  a.e.cae 34233

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5253  ax-nul 5263  ax-pow 5322  ax-pr 5389  ax-un 7713

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3756  df-csb 3865  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-nul 4299  df-if 4491  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5110  df-opab 5172  df-mpt 5191  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-iota 6466  df-fun 6515  df-fn 6516  df-f 6517  df-fv 6521  df-ov 7392  df-esum 34024  df-meas 34192  df-ae 34235

This theorem is referenced by: (None)