Theorem brfae 34432

Description: 'almost everywhere' relation for two functions 𝐹 and 𝐺 with regard to the measure 𝑀. (Contributed by Thierry Arnoux, 22-Oct-2017.)

Hypotheses

Ref	Expression
brfae.0	⊢ dom 𝑅 = 𝐷
brfae.1	⊢ (𝜑 → 𝑅 ∈ V)
brfae.2	⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)
brfae.3	⊢ (𝜑 → 𝐹 ∈ (𝐷 ↑m ∪ dom 𝑀))
brfae.4	⊢ (𝜑 → 𝐺 ∈ (𝐷 ↑m ∪ dom 𝑀))

Assertion

Ref	Expression
brfae	⊢ (𝜑 → (𝐹(𝑅~ a.e.𝑀)𝐺 ↔ {𝑥 ∈ ∪ dom 𝑀 ∣ (𝐹‘𝑥)𝑅(𝐺‘𝑥)}a.e.𝑀))

Distinct variable groups:   𝑥,𝐹   𝑥,𝐺   𝑥,𝑀   𝑥,𝑅

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

Proof of Theorem brfae

Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		brfae.3	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝐷 ↑m ∪ dom 𝑀))
2		brfae.4	. . 3 ⊢ (𝜑 → 𝐺 ∈ (𝐷 ↑m ∪ dom 𝑀))
3		simpl 483	. . . . . . 7 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → 𝑓 = 𝐹)
4	3	eleq1d 2824	. . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑓 ∈ (dom 𝑅 ↑m ∪ dom 𝑀) ↔ 𝐹 ∈ (dom 𝑅 ↑m ∪ dom 𝑀)))
5		simpr 485	. . . . . . 7 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → 𝑔 = 𝐺)
6	5	eleq1d 2824	. . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑔 ∈ (dom 𝑅 ↑m ∪ dom 𝑀) ↔ 𝐺 ∈ (dom 𝑅 ↑m ∪ dom 𝑀)))
7	4, 6	anbi12d 638	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ((𝑓 ∈ (dom 𝑅 ↑m ∪ dom 𝑀) ∧ 𝑔 ∈ (dom 𝑅 ↑m ∪ dom 𝑀)) ↔ (𝐹 ∈ (dom 𝑅 ↑m ∪ dom 𝑀) ∧ 𝐺 ∈ (dom 𝑅 ↑m ∪ dom 𝑀))))
8	3	fveq1d 6829	. . . . . . . 8 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑓‘𝑥) = (𝐹‘𝑥))
9	5	fveq1d 6829	. . . . . . . 8 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑔‘𝑥) = (𝐺‘𝑥))
10	8, 9	breq12d 5085	. . . . . . 7 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ((𝑓‘𝑥)𝑅(𝑔‘𝑥) ↔ (𝐹‘𝑥)𝑅(𝐺‘𝑥)))
11	10	rabbidv 3398	. . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → {𝑥 ∈ ∪ dom 𝑀 ∣ (𝑓‘𝑥)𝑅(𝑔‘𝑥)} = {𝑥 ∈ ∪ dom 𝑀 ∣ (𝐹‘𝑥)𝑅(𝐺‘𝑥)})
12	11	breq1d 5082	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ({𝑥 ∈ ∪ dom 𝑀 ∣ (𝑓‘𝑥)𝑅(𝑔‘𝑥)}a.e.𝑀 ↔ {𝑥 ∈ ∪ dom 𝑀 ∣ (𝐹‘𝑥)𝑅(𝐺‘𝑥)}a.e.𝑀))
13	7, 12	anbi12d 638	. . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (((𝑓 ∈ (dom 𝑅 ↑m ∪ dom 𝑀) ∧ 𝑔 ∈ (dom 𝑅 ↑m ∪ dom 𝑀)) ∧ {𝑥 ∈ ∪ dom 𝑀 ∣ (𝑓‘𝑥)𝑅(𝑔‘𝑥)}a.e.𝑀) ↔ ((𝐹 ∈ (dom 𝑅 ↑m ∪ dom 𝑀) ∧ 𝐺 ∈ (dom 𝑅 ↑m ∪ dom 𝑀)) ∧ {𝑥 ∈ ∪ dom 𝑀 ∣ (𝐹‘𝑥)𝑅(𝐺‘𝑥)}a.e.𝑀)))
14		eqid 2739	. . . 4 ⊢ {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (dom 𝑅 ↑m ∪ dom 𝑀) ∧ 𝑔 ∈ (dom 𝑅 ↑m ∪ dom 𝑀)) ∧ {𝑥 ∈ ∪ dom 𝑀 ∣ (𝑓‘𝑥)𝑅(𝑔‘𝑥)}a.e.𝑀)} = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (dom 𝑅 ↑m ∪ dom 𝑀) ∧ 𝑔 ∈ (dom 𝑅 ↑m ∪ dom 𝑀)) ∧ {𝑥 ∈ ∪ dom 𝑀 ∣ (𝑓‘𝑥)𝑅(𝑔‘𝑥)}a.e.𝑀)}
15	13, 14	brabga 5476	. . 3 ⊢ ((𝐹 ∈ (𝐷 ↑m ∪ dom 𝑀) ∧ 𝐺 ∈ (𝐷 ↑m ∪ dom 𝑀)) → (𝐹{⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (dom 𝑅 ↑m ∪ dom 𝑀) ∧ 𝑔 ∈ (dom 𝑅 ↑m ∪ dom 𝑀)) ∧ {𝑥 ∈ ∪ dom 𝑀 ∣ (𝑓‘𝑥)𝑅(𝑔‘𝑥)}a.e.𝑀)}𝐺 ↔ ((𝐹 ∈ (dom 𝑅 ↑m ∪ dom 𝑀) ∧ 𝐺 ∈ (dom 𝑅 ↑m ∪ dom 𝑀)) ∧ {𝑥 ∈ ∪ dom 𝑀 ∣ (𝐹‘𝑥)𝑅(𝐺‘𝑥)}a.e.𝑀)))
16	1, 2, 15	syl2anc 590	. 2 ⊢ (𝜑 → (𝐹{⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (dom 𝑅 ↑m ∪ dom 𝑀) ∧ 𝑔 ∈ (dom 𝑅 ↑m ∪ dom 𝑀)) ∧ {𝑥 ∈ ∪ dom 𝑀 ∣ (𝑓‘𝑥)𝑅(𝑔‘𝑥)}a.e.𝑀)}𝐺 ↔ ((𝐹 ∈ (dom 𝑅 ↑m ∪ dom 𝑀) ∧ 𝐺 ∈ (dom 𝑅 ↑m ∪ dom 𝑀)) ∧ {𝑥 ∈ ∪ dom 𝑀 ∣ (𝐹‘𝑥)𝑅(𝐺‘𝑥)}a.e.𝑀)))
17		brfae.1	. . . 4 ⊢ (𝜑 → 𝑅 ∈ V)
18		brfae.2	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)
19		faeval 34430	. . . 4 ⊢ ((𝑅 ∈ V ∧ 𝑀 ∈ ∪ ran measures) → (𝑅~ a.e.𝑀) = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (dom 𝑅 ↑m ∪ dom 𝑀) ∧ 𝑔 ∈ (dom 𝑅 ↑m ∪ dom 𝑀)) ∧ {𝑥 ∈ ∪ dom 𝑀 ∣ (𝑓‘𝑥)𝑅(𝑔‘𝑥)}a.e.𝑀)})
20	17, 18, 19	syl2anc 590	. . 3 ⊢ (𝜑 → (𝑅~ a.e.𝑀) = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (dom 𝑅 ↑m ∪ dom 𝑀) ∧ 𝑔 ∈ (dom 𝑅 ↑m ∪ dom 𝑀)) ∧ {𝑥 ∈ ∪ dom 𝑀 ∣ (𝑓‘𝑥)𝑅(𝑔‘𝑥)}a.e.𝑀)})
21	20	breqd 5083	. 2 ⊢ (𝜑 → (𝐹(𝑅~ a.e.𝑀)𝐺 ↔ 𝐹{⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (dom 𝑅 ↑m ∪ dom 𝑀) ∧ 𝑔 ∈ (dom 𝑅 ↑m ∪ dom 𝑀)) ∧ {𝑥 ∈ ∪ dom 𝑀 ∣ (𝑓‘𝑥)𝑅(𝑔‘𝑥)}a.e.𝑀)}𝐺))
22		brfae.0	. . . . . 6 ⊢ dom 𝑅 = 𝐷
23	22	oveq1i 7366	. . . . 5 ⊢ (dom 𝑅 ↑m ∪ dom 𝑀) = (𝐷 ↑m ∪ dom 𝑀)
24	1, 23	eleqtrrdi 2850	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (dom 𝑅 ↑m ∪ dom 𝑀))
25	2, 23	eleqtrrdi 2850	. . . 4 ⊢ (𝜑 → 𝐺 ∈ (dom 𝑅 ↑m ∪ dom 𝑀))
26	24, 25	jca 516	. . 3 ⊢ (𝜑 → (𝐹 ∈ (dom 𝑅 ↑m ∪ dom 𝑀) ∧ 𝐺 ∈ (dom 𝑅 ↑m ∪ dom 𝑀)))
27	26	biantrurd 537	. 2 ⊢ (𝜑 → ({𝑥 ∈ ∪ dom 𝑀 ∣ (𝐹‘𝑥)𝑅(𝐺‘𝑥)}a.e.𝑀 ↔ ((𝐹 ∈ (dom 𝑅 ↑m ∪ dom 𝑀) ∧ 𝐺 ∈ (dom 𝑅 ↑m ∪ dom 𝑀)) ∧ {𝑥 ∈ ∪ dom 𝑀 ∣ (𝐹‘𝑥)𝑅(𝐺‘𝑥)}a.e.𝑀)))
28	16, 21, 27	3bitr4d 312	1 ⊢ (𝜑 → (𝐹(𝑅~ a.e.𝑀)𝐺 ↔ {𝑥 ∈ ∪ dom 𝑀 ∣ (𝐹‘𝑥)𝑅(𝐺‘𝑥)}a.e.𝑀))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119  {crab 3391  Vcvv 3431  ∪ cuni 4838   class class class wbr 5072  {copab 5134  dom cdm 5618  ran crn 5619  ‘cfv 6485  (class class class)co 7356   ↑_m cmap 8763  measurescmeas 34379  a.e.cae 34421  ~ a.e.cfae 34422

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-sbc 3724  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-iota 6441  df-fun 6487  df-fv 6493  df-ov 7359  df-oprab 7360  df-mpo 7361  df-fae 34429

This theorem is referenced by: (None)