Theorem braew 31611

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 7594	. . . . . 6 ⊢ (𝑀 ∈ ∪ ran measures → dom 𝑀 ∈ V)
3	2	uniexd 7448	. . . . 5 ⊢ (𝑀 ∈ ∪ ran measures → ∪ dom 𝑀 ∈ V)
4	1, 3	eqeltrrid 2895	. . . 4 ⊢ (𝑀 ∈ ∪ ran measures → 𝑂 ∈ V)
5		rabexg 5198	. . . 4 ⊢ (𝑂 ∈ V → {𝑥 ∈ 𝑂 ∣ 𝜑} ∈ V)
6	4, 5	syl 17	. . 3 ⊢ (𝑀 ∈ ∪ ran measures → {𝑥 ∈ 𝑂 ∣ 𝜑} ∈ V)
7		simpr 488	. . . . . 6 ⊢ ((𝑎 = {𝑥 ∈ 𝑂 ∣ 𝜑} ∧ 𝑚 = 𝑀) → 𝑚 = 𝑀)
8	7	dmeqd 5738	. . . . . . . 8 ⊢ ((𝑎 = {𝑥 ∈ 𝑂 ∣ 𝜑} ∧ 𝑚 = 𝑀) → dom 𝑚 = dom 𝑀)
9	8	unieqd 4814	. . . . . . 7 ⊢ ((𝑎 = {𝑥 ∈ 𝑂 ∣ 𝜑} ∧ 𝑚 = 𝑀) → ∪ dom 𝑚 = ∪ dom 𝑀)
10		simpl 486	. . . . . . 7 ⊢ ((𝑎 = {𝑥 ∈ 𝑂 ∣ 𝜑} ∧ 𝑚 = 𝑀) → 𝑎 = {𝑥 ∈ 𝑂 ∣ 𝜑})
11	9, 10	difeq12d 4051	. . . . . 6 ⊢ ((𝑎 = {𝑥 ∈ 𝑂 ∣ 𝜑} ∧ 𝑚 = 𝑀) → (∪ dom 𝑚 ∖ 𝑎) = (∪ dom 𝑀 ∖ {𝑥 ∈ 𝑂 ∣ 𝜑}))
12	7, 11	fveq12d 6652	. . . . 5 ⊢ ((𝑎 = {𝑥 ∈ 𝑂 ∣ 𝜑} ∧ 𝑚 = 𝑀) → (𝑚‘(∪ dom 𝑚 ∖ 𝑎)) = (𝑀‘(∪ dom 𝑀 ∖ {𝑥 ∈ 𝑂 ∣ 𝜑})))
13	12	eqeq1d 2800	. . . 4 ⊢ ((𝑎 = {𝑥 ∈ 𝑂 ∣ 𝜑} ∧ 𝑚 = 𝑀) → ((𝑚‘(∪ dom 𝑚 ∖ 𝑎)) = 0 ↔ (𝑀‘(∪ dom 𝑀 ∖ {𝑥 ∈ 𝑂 ∣ 𝜑})) = 0))
14		df-ae 31608	. . . 4 ⊢ a.e. = {⟨𝑎, 𝑚⟩ ∣ (𝑚‘(∪ dom 𝑚 ∖ 𝑎)) = 0}
15	13, 14	brabga 5386	. . 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 687	. 2 ⊢ (𝑀 ∈ ∪ ran measures → ({𝑥 ∈ 𝑂 ∣ 𝜑}a.e.𝑀 ↔ (𝑀‘(∪ dom 𝑀 ∖ {𝑥 ∈ 𝑂 ∣ 𝜑})) = 0))
17	1	difeq1i 4046	. . . . 5 ⊢ (∪ dom 𝑀 ∖ {𝑥 ∈ 𝑂 ∣ 𝜑}) = (𝑂 ∖ {𝑥 ∈ 𝑂 ∣ 𝜑})
18		notrab 4232	. . . . 5 ⊢ (𝑂 ∖ {𝑥 ∈ 𝑂 ∣ 𝜑}) = {𝑥 ∈ 𝑂 ∣ ¬ 𝜑}
19	17, 18	eqtri 2821	. . . 4 ⊢ (∪ dom 𝑀 ∖ {𝑥 ∈ 𝑂 ∣ 𝜑}) = {𝑥 ∈ 𝑂 ∣ ¬ 𝜑}
20	19	fveq2i 6648	. . 3 ⊢ (𝑀‘(∪ dom 𝑀 ∖ {𝑥 ∈ 𝑂 ∣ 𝜑})) = (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜑})
21	20	eqeq1i 2803	. 2 ⊢ ((𝑀‘(∪ dom 𝑀 ∖ {𝑥 ∈ 𝑂 ∣ 𝜑})) = 0 ↔ (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜑}) = 0)
22	16, 21	syl6bb 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 399   = wceq 1538   ∈ wcel 2111  {crab 3110  Vcvv 3441   ∖ cdif 3878  ∪ cuni 4800   class class class wbr 5030  dom cdm 5519  ran crn 5520  ‘cfv 6324  0cc0 10526  measurescmeas 31564  a.e.cae 31606

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-rab 3115  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-cnv 5527  df-dm 5529  df-rn 5530  df-iota 6283  df-fv 6332  df-ae 31608

This theorem is referenced by:  truae  31612  aean  31613