Theorem truae 34256

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 3128	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝑂 (¬ 𝜓 → 𝑥 ∈ ∅))
4		rabss 4017	. . . . . 6 ⊢ ({𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ⊆ ∅ ↔ ∀𝑥 ∈ 𝑂 (¬ 𝜓 → 𝑥 ∈ ∅))
5	3, 4	sylibr 234	. . . . 5 ⊢ (𝜑 → {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ⊆ ∅)
6		ss0 4349	. . . . 5 ⊢ ({𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ⊆ ∅ → {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} = ∅)
7	5, 6	syl 17	. . . 4 ⊢ (𝜑 → {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} = ∅)
8	7	fveq2d 6826	. . 3 ⊢ (𝜑 → (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜓}) = (𝑀‘∅))
9		truae.2	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)
10		measbasedom 34215	. . . . 5 ⊢ (𝑀 ∈ ∪ ran measures ↔ 𝑀 ∈ (measures‘dom 𝑀))
11		measvnul 34219	. . . . 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 2766	. 2 ⊢ (𝜑 → (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜓}) = 0)
15		truae.1	. . . 4 ⊢ ∪ dom 𝑀 = 𝑂
16	15	braew 34255	. . 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 1541   ∈ wcel 2111  ∀wral 3047  {crab 3395   ⊆ wss 3897  ∅c0 4280  ∪ cuni 4856   class class class wbr 5089  dom cdm 5614  ran crn 5615  ‘cfv 6481  0cc0 11006  measurescmeas 34208  a.e.cae 34250

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-fv 6489  df-ov 7349  df-esum 34041  df-meas 34209  df-ae 34252

This theorem is referenced by: (None)