Theorem braew 34544

Description: 'almost everywhere' relation for a measure 𝑀 and a property 𝜑 (Contributed by Thierry Arnoux, 20-Oct-2017.)

Hypothesis

Ref	Expression
braew.1	⊢ ∪ dom 𝑀 = 𝑂

Assertion

Ref	Expression
braew	⊢ (𝑀 ∈ ∪ ran measures → ({𝑥 ∈ 𝑂 ∣ 𝜑}a.e.𝑀 ↔ (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜑}) = 0))

Distinct variable group:   𝑥,𝑂

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

Proof of Theorem braew

Dummy variables 𝑚 𝑎 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		braew.1	. . . . 5 ⊢ ∪ dom 𝑀 = 𝑂
2		dmexg 7886	. . . . . 6 ⊢ (𝑀 ∈ ∪ ran measures → dom 𝑀 ∈ V)
3	2	uniexd 7729	. . . . 5 ⊢ (𝑀 ∈ ∪ ran measures → ∪ dom 𝑀 ∈ V)
4	1, 3	eqeltrrid 2870	. . . 4 ⊢ (𝑀 ∈ ∪ ran measures → 𝑂 ∈ V)
5		rabexg 5297	. . . 4 ⊢ (𝑂 ∈ V → {𝑥 ∈ 𝑂 ∣ 𝜑} ∈ V)
6	4, 5	syl 18	. . 3 ⊢ (𝑀 ∈ ∪ ran measures → {𝑥 ∈ 𝑂 ∣ 𝜑} ∈ V)
7		simpr 489	. . . . . 6 ⊢ ((𝑎 = {𝑥 ∈ 𝑂 ∣ 𝜑} ∧ 𝑚 = 𝑀) → 𝑚 = 𝑀)
8	7	dmeqd 5885	. . . . . . . 8 ⊢ ((𝑎 = {𝑥 ∈ 𝑂 ∣ 𝜑} ∧ 𝑚 = 𝑀) → dom 𝑚 = dom 𝑀)
9	8	unieqd 4880	. . . . . . 7 ⊢ ((𝑎 = {𝑥 ∈ 𝑂 ∣ 𝜑} ∧ 𝑚 = 𝑀) → ∪ dom 𝑚 = ∪ dom 𝑀)
10		simpl 487	. . . . . . 7 ⊢ ((𝑎 = {𝑥 ∈ 𝑂 ∣ 𝜑} ∧ 𝑚 = 𝑀) → 𝑎 = {𝑥 ∈ 𝑂 ∣ 𝜑})
11	9, 10	difeq12d 4084	. . . . . 6 ⊢ ((𝑎 = {𝑥 ∈ 𝑂 ∣ 𝜑} ∧ 𝑚 = 𝑀) → (∪ dom 𝑚 ∖ 𝑎) = (∪ dom 𝑀 ∖ {𝑥 ∈ 𝑂 ∣ 𝜑}))
12	7, 11	fveq12d 6878	. . . . 5 ⊢ ((𝑎 = {𝑥 ∈ 𝑂 ∣ 𝜑} ∧ 𝑚 = 𝑀) → (𝑚‘(∪ dom 𝑚 ∖ 𝑎)) = (𝑀‘(∪ dom 𝑀 ∖ {𝑥 ∈ 𝑂 ∣ 𝜑})))
13	12	eqeq1d 2767	. . . 4 ⊢ ((𝑎 = {𝑥 ∈ 𝑂 ∣ 𝜑} ∧ 𝑚 = 𝑀) → ((𝑚‘(∪ dom 𝑚 ∖ 𝑎)) = 0 ↔ (𝑀‘(∪ dom 𝑀 ∖ {𝑥 ∈ 𝑂 ∣ 𝜑})) = 0))
14		df-ae 34541	. . . 4 ⊢ a.e. = {⟨𝑎, 𝑚⟩ ∣ (𝑚‘(∪ dom 𝑚 ∖ 𝑎)) = 0}
15	13, 14	brabga 5508	. . 3 ⊢ (({𝑥 ∈ 𝑂 ∣ 𝜑} ∈ V ∧ 𝑀 ∈ ∪ ran measures) → ({𝑥 ∈ 𝑂 ∣ 𝜑}a.e.𝑀 ↔ (𝑀‘(∪ dom 𝑀 ∖ {𝑥 ∈ 𝑂 ∣ 𝜑})) = 0))
16	6, 15	mpancom 700	. 2 ⊢ (𝑀 ∈ ∪ ran measures → ({𝑥 ∈ 𝑂 ∣ 𝜑}a.e.𝑀 ↔ (𝑀‘(∪ dom 𝑀 ∖ {𝑥 ∈ 𝑂 ∣ 𝜑})) = 0))
17	1	difeq1i 4079	. . . . 5 ⊢ (∪ dom 𝑀 ∖ {𝑥 ∈ 𝑂 ∣ 𝜑}) = (𝑂 ∖ {𝑥 ∈ 𝑂 ∣ 𝜑})
18		notrab 4277	. . . . 5 ⊢ (𝑂 ∖ {𝑥 ∈ 𝑂 ∣ 𝜑}) = {𝑥 ∈ 𝑂 ∣ ¬ 𝜑}
19	17, 18	eqtri 2788	. . . 4 ⊢ (∪ dom 𝑀 ∖ {𝑥 ∈ 𝑂 ∣ 𝜑}) = {𝑥 ∈ 𝑂 ∣ ¬ 𝜑}
20	19	fveq2i 6874	. . 3 ⊢ (𝑀‘(∪ dom 𝑀 ∖ {𝑥 ∈ 𝑂 ∣ 𝜑})) = (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜑})
21	20	eqeq1i 2770	. 2 ⊢ ((𝑀‘(∪ dom 𝑀 ∖ {𝑥 ∈ 𝑂 ∣ 𝜑})) = 0 ↔ (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜑}) = 0)
22	16, 21	bitrdi 290	1 ⊢ (𝑀 ∈ ∪ ran measures → ({𝑥 ∈ 𝑂 ∣ 𝜑}a.e.𝑀 ↔ (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜑}) = 0))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1563   ∈ wcel 2145  {crab 3417  Vcvv 3457   ∖ cdif 3904  ∪ cuni 4867   class class class wbr 5104  dom cdm 5651  ran crn 5652  ‘cfv 6525  0cc0 11088  measurescmeas 34497  a.e.cae 34539

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-12 2215  ax-ext 2737  ax-sep 5250  ax-pr 5394  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5105  df-opab 5167  df-cnv 5659  df-dm 5661  df-rn 5662  df-iota 6481  df-fv 6533  df-ae 34541

This theorem is referenced by:  truae  34545  aean  34546