Theorem braew 34399

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 7843	. . . . . 6 ⊢ (𝑀 ∈ ∪ ran measures → dom 𝑀 ∈ V)
3	2	uniexd 7687	. . . . 5 ⊢ (𝑀 ∈ ∪ ran measures → ∪ dom 𝑀 ∈ V)
4	1, 3	eqeltrrid 2841	. . . 4 ⊢ (𝑀 ∈ ∪ ran measures → 𝑂 ∈ V)
5		rabexg 5282	. . . 4 ⊢ (𝑂 ∈ V → {𝑥 ∈ 𝑂 ∣ 𝜑} ∈ V)
6	4, 5	syl 17	. . 3 ⊢ (𝑀 ∈ ∪ ran measures → {𝑥 ∈ 𝑂 ∣ 𝜑} ∈ V)
7		simpr 484	. . . . . 6 ⊢ ((𝑎 = {𝑥 ∈ 𝑂 ∣ 𝜑} ∧ 𝑚 = 𝑀) → 𝑚 = 𝑀)
8	7	dmeqd 5854	. . . . . . . 8 ⊢ ((𝑎 = {𝑥 ∈ 𝑂 ∣ 𝜑} ∧ 𝑚 = 𝑀) → dom 𝑚 = dom 𝑀)
9	8	unieqd 4876	. . . . . . 7 ⊢ ((𝑎 = {𝑥 ∈ 𝑂 ∣ 𝜑} ∧ 𝑚 = 𝑀) → ∪ dom 𝑚 = ∪ dom 𝑀)
10		simpl 482	. . . . . . 7 ⊢ ((𝑎 = {𝑥 ∈ 𝑂 ∣ 𝜑} ∧ 𝑚 = 𝑀) → 𝑎 = {𝑥 ∈ 𝑂 ∣ 𝜑})
11	9, 10	difeq12d 4079	. . . . . 6 ⊢ ((𝑎 = {𝑥 ∈ 𝑂 ∣ 𝜑} ∧ 𝑚 = 𝑀) → (∪ dom 𝑚 ∖ 𝑎) = (∪ dom 𝑀 ∖ {𝑥 ∈ 𝑂 ∣ 𝜑}))
12	7, 11	fveq12d 6841	. . . . 5 ⊢ ((𝑎 = {𝑥 ∈ 𝑂 ∣ 𝜑} ∧ 𝑚 = 𝑀) → (𝑚‘(∪ dom 𝑚 ∖ 𝑎)) = (𝑀‘(∪ dom 𝑀 ∖ {𝑥 ∈ 𝑂 ∣ 𝜑})))
13	12	eqeq1d 2738	. . . 4 ⊢ ((𝑎 = {𝑥 ∈ 𝑂 ∣ 𝜑} ∧ 𝑚 = 𝑀) → ((𝑚‘(∪ dom 𝑚 ∖ 𝑎)) = 0 ↔ (𝑀‘(∪ dom 𝑀 ∖ {𝑥 ∈ 𝑂 ∣ 𝜑})) = 0))
14		df-ae 34396	. . . 4 ⊢ a.e. = {⟨𝑎, 𝑚⟩ ∣ (𝑚‘(∪ dom 𝑚 ∖ 𝑎)) = 0}
15	13, 14	brabga 5482	. . 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 688	. 2 ⊢ (𝑀 ∈ ∪ ran measures → ({𝑥 ∈ 𝑂 ∣ 𝜑}a.e.𝑀 ↔ (𝑀‘(∪ dom 𝑀 ∖ {𝑥 ∈ 𝑂 ∣ 𝜑})) = 0))
17	1	difeq1i 4074	. . . . 5 ⊢ (∪ dom 𝑀 ∖ {𝑥 ∈ 𝑂 ∣ 𝜑}) = (𝑂 ∖ {𝑥 ∈ 𝑂 ∣ 𝜑})
18		notrab 4274	. . . . 5 ⊢ (𝑂 ∖ {𝑥 ∈ 𝑂 ∣ 𝜑}) = {𝑥 ∈ 𝑂 ∣ ¬ 𝜑}
19	17, 18	eqtri 2759	. . . 4 ⊢ (∪ dom 𝑀 ∖ {𝑥 ∈ 𝑂 ∣ 𝜑}) = {𝑥 ∈ 𝑂 ∣ ¬ 𝜑}
20	19	fveq2i 6837	. . 3 ⊢ (𝑀‘(∪ dom 𝑀 ∖ {𝑥 ∈ 𝑂 ∣ 𝜑})) = (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜑})
21	20	eqeq1i 2741	. 2 ⊢ ((𝑀‘(∪ dom 𝑀 ∖ {𝑥 ∈ 𝑂 ∣ 𝜑})) = 0 ↔ (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜑}) = 0)
22	16, 21	bitrdi 287	1 ⊢ (𝑀 ∈ ∪ ran measures → ({𝑥 ∈ 𝑂 ∣ 𝜑}a.e.𝑀 ↔ (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜑}) = 0))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  {crab 3399  Vcvv 3440   ∖ cdif 3898  ∪ cuni 4863   class class class wbr 5098  dom cdm 5624  ran crn 5625  ‘cfv 6492  0cc0 11026  measurescmeas 34352  a.e.cae 34394

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 2115  ax-9 2123  ax-10 2146  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680

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-clab 2715  df-cleq 2728  df-clel 2811  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-cnv 5632  df-dm 5634  df-rn 5635  df-iota 6448  df-fv 6500  df-ae 34396

This theorem is referenced by:  truae  34400  aean  34401