Theorem braew 30614

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 7262	. . . . . 6 ⊢ (𝑀 ∈ ∪ ran measures → dom 𝑀 ∈ V)
3		uniexg 7120	. . . . . 6 ⊢ (dom 𝑀 ∈ V → ∪ dom 𝑀 ∈ V)
4	2, 3	syl 17	. . . . 5 ⊢ (𝑀 ∈ ∪ ran measures → ∪ dom 𝑀 ∈ V)
5	1, 4	syl5eqelr 2844	. . . 4 ⊢ (𝑀 ∈ ∪ ran measures → 𝑂 ∈ V)
6		rabexg 4963	. . . 4 ⊢ (𝑂 ∈ V → {𝑥 ∈ 𝑂 ∣ 𝜑} ∈ V)
7	5, 6	syl 17	. . 3 ⊢ (𝑀 ∈ ∪ ran measures → {𝑥 ∈ 𝑂 ∣ 𝜑} ∈ V)
8		simpr 479	. . . . . 6 ⊢ ((𝑎 = {𝑥 ∈ 𝑂 ∣ 𝜑} ∧ 𝑚 = 𝑀) → 𝑚 = 𝑀)
9	8	dmeqd 5481	. . . . . . . 8 ⊢ ((𝑎 = {𝑥 ∈ 𝑂 ∣ 𝜑} ∧ 𝑚 = 𝑀) → dom 𝑚 = dom 𝑀)
10	9	unieqd 4598	. . . . . . 7 ⊢ ((𝑎 = {𝑥 ∈ 𝑂 ∣ 𝜑} ∧ 𝑚 = 𝑀) → ∪ dom 𝑚 = ∪ dom 𝑀)
11		simpl 474	. . . . . . 7 ⊢ ((𝑎 = {𝑥 ∈ 𝑂 ∣ 𝜑} ∧ 𝑚 = 𝑀) → 𝑎 = {𝑥 ∈ 𝑂 ∣ 𝜑})
12	10, 11	difeq12d 3872	. . . . . 6 ⊢ ((𝑎 = {𝑥 ∈ 𝑂 ∣ 𝜑} ∧ 𝑚 = 𝑀) → (∪ dom 𝑚 ∖ 𝑎) = (∪ dom 𝑀 ∖ {𝑥 ∈ 𝑂 ∣ 𝜑}))
13	8, 12	fveq12d 6358	. . . . 5 ⊢ ((𝑎 = {𝑥 ∈ 𝑂 ∣ 𝜑} ∧ 𝑚 = 𝑀) → (𝑚‘(∪ dom 𝑚 ∖ 𝑎)) = (𝑀‘(∪ dom 𝑀 ∖ {𝑥 ∈ 𝑂 ∣ 𝜑})))
14	13	eqeq1d 2762	. . . 4 ⊢ ((𝑎 = {𝑥 ∈ 𝑂 ∣ 𝜑} ∧ 𝑚 = 𝑀) → ((𝑚‘(∪ dom 𝑚 ∖ 𝑎)) = 0 ↔ (𝑀‘(∪ dom 𝑀 ∖ {𝑥 ∈ 𝑂 ∣ 𝜑})) = 0))
15		df-ae 30611	. . . 4 ⊢ a.e. = {⟨𝑎, 𝑚⟩ ∣ (𝑚‘(∪ dom 𝑚 ∖ 𝑎)) = 0}
16	14, 15	brabga 5139	. . 3 ⊢ (({𝑥 ∈ 𝑂 ∣ 𝜑} ∈ V ∧ 𝑀 ∈ ∪ ran measures) → ({𝑥 ∈ 𝑂 ∣ 𝜑}a.e.𝑀 ↔ (𝑀‘(∪ dom 𝑀 ∖ {𝑥 ∈ 𝑂 ∣ 𝜑})) = 0))
17	7, 16	mpancom 706	. 2 ⊢ (𝑀 ∈ ∪ ran measures → ({𝑥 ∈ 𝑂 ∣ 𝜑}a.e.𝑀 ↔ (𝑀‘(∪ dom 𝑀 ∖ {𝑥 ∈ 𝑂 ∣ 𝜑})) = 0))
18	1	difeq1i 3867	. . . . 5 ⊢ (∪ dom 𝑀 ∖ {𝑥 ∈ 𝑂 ∣ 𝜑}) = (𝑂 ∖ {𝑥 ∈ 𝑂 ∣ 𝜑})
19		notrab 4047	. . . . 5 ⊢ (𝑂 ∖ {𝑥 ∈ 𝑂 ∣ 𝜑}) = {𝑥 ∈ 𝑂 ∣ ¬ 𝜑}
20	18, 19	eqtri 2782	. . . 4 ⊢ (∪ dom 𝑀 ∖ {𝑥 ∈ 𝑂 ∣ 𝜑}) = {𝑥 ∈ 𝑂 ∣ ¬ 𝜑}
21	20	fveq2i 6355	. . 3 ⊢ (𝑀‘(∪ dom 𝑀 ∖ {𝑥 ∈ 𝑂 ∣ 𝜑})) = (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜑})
22	21	eqeq1i 2765	. 2 ⊢ ((𝑀‘(∪ dom 𝑀 ∖ {𝑥 ∈ 𝑂 ∣ 𝜑})) = 0 ↔ (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜑}) = 0)
23	17, 22	syl6bb 276	1 ⊢ (𝑀 ∈ ∪ ran measures → ({𝑥 ∈ 𝑂 ∣ 𝜑}a.e.𝑀 ↔ (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜑}) = 0))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1632   ∈ wcel 2139  {crab 3054  Vcvv 3340   ∖ cdif 3712  ∪ cuni 4588   class class class wbr 4804  dom cdm 5266  ran crn 5267  ‘cfv 6049  0cc0 10128  measurescmeas 30567  a.e.cae 30609

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055  ax-un 7114

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-cnv 5274  df-dm 5276  df-rn 5277  df-iota 6012  df-fv 6057  df-ae 30611

This theorem is referenced by:  truae  30615  aean  30616