Theorem truae 34425

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 3134	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝑂 (¬ 𝜓 → 𝑥 ∈ ∅))
4		rabss 4024	. . . . . 6 ⊢ ({𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ⊆ ∅ ↔ ∀𝑥 ∈ 𝑂 (¬ 𝜓 → 𝑥 ∈ ∅))
5	3, 4	sylibr 234	. . . . 5 ⊢ (𝜑 → {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ⊆ ∅)
6		ss0 4356	. . . . 5 ⊢ ({𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ⊆ ∅ → {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} = ∅)
7	5, 6	syl 17	. . . 4 ⊢ (𝜑 → {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} = ∅)
8	7	fveq2d 6846	. . 3 ⊢ (𝜑 → (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜓}) = (𝑀‘∅))
9		truae.2	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)
10		measbasedom 34384	. . . . 5 ⊢ (𝑀 ∈ ∪ ran measures ↔ 𝑀 ∈ (measures‘dom 𝑀))
11		measvnul 34388	. . . . 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 2772	. 2 ⊢ (𝜑 → (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜓}) = 0)
15		truae.1	. . . 4 ⊢ ∪ dom 𝑀 = 𝑂
16	15	braew 34424	. . 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 1542   ∈ wcel 2114  ∀wral 3052  {crab 3401   ⊆ wss 3903  ∅c0 4287  ∪ cuni 4865   class class class wbr 5100  dom cdm 5632  ran crn 5633  ‘cfv 6500  0cc0 11038  measurescmeas 34377  a.e.cae 34419

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-fv 6508  df-ov 7371  df-esum 34210  df-meas 34378  df-ae 34421

This theorem is referenced by: (None)