Theorem truae 33798

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 3145	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝑂 (¬ 𝜓 → 𝑥 ∈ ∅))
4		rabss 4065	. . . . . 6 ⊢ ({𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ⊆ ∅ ↔ ∀𝑥 ∈ 𝑂 (¬ 𝜓 → 𝑥 ∈ ∅))
5	3, 4	sylibr 233	. . . . 5 ⊢ (𝜑 → {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ⊆ ∅)
6		ss0 4394	. . . . 5 ⊢ ({𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ⊆ ∅ → {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} = ∅)
7	5, 6	syl 17	. . . 4 ⊢ (𝜑 → {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} = ∅)
8	7	fveq2d 6895	. . 3 ⊢ (𝜑 → (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜓}) = (𝑀‘∅))
9		truae.2	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)
10		measbasedom 33757	. . . . 5 ⊢ (𝑀 ∈ ∪ ran measures ↔ 𝑀 ∈ (measures‘dom 𝑀))
11		measvnul 33761	. . . . 5 ⊢ (𝑀 ∈ (measures‘dom 𝑀) → (𝑀‘∅) = 0)
12	10, 11	sylbi 216	. . . 4 ⊢ (𝑀 ∈ ∪ ran measures → (𝑀‘∅) = 0)
13	9, 12	syl 17	. . 3 ⊢ (𝜑 → (𝑀‘∅) = 0)
14	8, 13	eqtrd 2767	. 2 ⊢ (𝜑 → (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜓}) = 0)
15		truae.1	. . . 4 ⊢ ∪ dom 𝑀 = 𝑂
16	15	braew 33797	. . 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 205   = wceq 1534   ∈ wcel 2099  ∀wral 3056  {crab 3427   ⊆ wss 3944  ∅c0 4318  ∪ cuni 4903   class class class wbr 5142  dom cdm 5672  ran crn 5673  ‘cfv 6542  0cc0 11130  measurescmeas 33750  a.e.cae 33792

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fv 6550  df-ov 7417  df-esum 33583  df-meas 33751  df-ae 33794

This theorem is referenced by: (None)