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Theorem brfae 33867
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 (𝜑𝑅 ∈ V)
brfae.2 (𝜑𝑀 ran measures)
brfae.3 (𝜑𝐹 ∈ (𝐷m dom 𝑀))
brfae.4 (𝜑𝐺 ∈ (𝐷m dom 𝑀))
Assertion
Ref Expression
brfae (𝜑 → (𝐹(𝑅~ a.e.𝑀)𝐺 ↔ {𝑥 dom 𝑀 ∣ (𝐹𝑥)𝑅(𝐺𝑥)}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.
StepHypRef Expression
1 brfae.3 . . 3 (𝜑𝐹 ∈ (𝐷m dom 𝑀))
2 brfae.4 . . 3 (𝜑𝐺 ∈ (𝐷m dom 𝑀))
3 simpl 482 . . . . . . 7 ((𝑓 = 𝐹𝑔 = 𝐺) → 𝑓 = 𝐹)
43eleq1d 2814 . . . . . 6 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑓 ∈ (dom 𝑅m dom 𝑀) ↔ 𝐹 ∈ (dom 𝑅m dom 𝑀)))
5 simpr 484 . . . . . . 7 ((𝑓 = 𝐹𝑔 = 𝐺) → 𝑔 = 𝐺)
65eleq1d 2814 . . . . . 6 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑔 ∈ (dom 𝑅m dom 𝑀) ↔ 𝐺 ∈ (dom 𝑅m dom 𝑀)))
74, 6anbi12d 631 . . . . 5 ((𝑓 = 𝐹𝑔 = 𝐺) → ((𝑓 ∈ (dom 𝑅m dom 𝑀) ∧ 𝑔 ∈ (dom 𝑅m dom 𝑀)) ↔ (𝐹 ∈ (dom 𝑅m dom 𝑀) ∧ 𝐺 ∈ (dom 𝑅m dom 𝑀))))
83fveq1d 6899 . . . . . . . 8 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑓𝑥) = (𝐹𝑥))
95fveq1d 6899 . . . . . . . 8 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑔𝑥) = (𝐺𝑥))
108, 9breq12d 5161 . . . . . . 7 ((𝑓 = 𝐹𝑔 = 𝐺) → ((𝑓𝑥)𝑅(𝑔𝑥) ↔ (𝐹𝑥)𝑅(𝐺𝑥)))
1110rabbidv 3437 . . . . . 6 ((𝑓 = 𝐹𝑔 = 𝐺) → {𝑥 dom 𝑀 ∣ (𝑓𝑥)𝑅(𝑔𝑥)} = {𝑥 dom 𝑀 ∣ (𝐹𝑥)𝑅(𝐺𝑥)})
1211breq1d 5158 . . . . 5 ((𝑓 = 𝐹𝑔 = 𝐺) → ({𝑥 dom 𝑀 ∣ (𝑓𝑥)𝑅(𝑔𝑥)}a.e.𝑀 ↔ {𝑥 dom 𝑀 ∣ (𝐹𝑥)𝑅(𝐺𝑥)}a.e.𝑀))
137, 12anbi12d 631 . . . 4 ((𝑓 = 𝐹𝑔 = 𝐺) → (((𝑓 ∈ (dom 𝑅m dom 𝑀) ∧ 𝑔 ∈ (dom 𝑅m dom 𝑀)) ∧ {𝑥 dom 𝑀 ∣ (𝑓𝑥)𝑅(𝑔𝑥)}a.e.𝑀) ↔ ((𝐹 ∈ (dom 𝑅m dom 𝑀) ∧ 𝐺 ∈ (dom 𝑅m dom 𝑀)) ∧ {𝑥 dom 𝑀 ∣ (𝐹𝑥)𝑅(𝐺𝑥)}a.e.𝑀)))
14 eqid 2728 . . . 4 {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (dom 𝑅m dom 𝑀) ∧ 𝑔 ∈ (dom 𝑅m dom 𝑀)) ∧ {𝑥 dom 𝑀 ∣ (𝑓𝑥)𝑅(𝑔𝑥)}a.e.𝑀)} = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (dom 𝑅m dom 𝑀) ∧ 𝑔 ∈ (dom 𝑅m dom 𝑀)) ∧ {𝑥 dom 𝑀 ∣ (𝑓𝑥)𝑅(𝑔𝑥)}a.e.𝑀)}
1513, 14brabga 5536 . . 3 ((𝐹 ∈ (𝐷m dom 𝑀) ∧ 𝐺 ∈ (𝐷m dom 𝑀)) → (𝐹{⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (dom 𝑅m dom 𝑀) ∧ 𝑔 ∈ (dom 𝑅m dom 𝑀)) ∧ {𝑥 dom 𝑀 ∣ (𝑓𝑥)𝑅(𝑔𝑥)}a.e.𝑀)}𝐺 ↔ ((𝐹 ∈ (dom 𝑅m dom 𝑀) ∧ 𝐺 ∈ (dom 𝑅m dom 𝑀)) ∧ {𝑥 dom 𝑀 ∣ (𝐹𝑥)𝑅(𝐺𝑥)}a.e.𝑀)))
161, 2, 15syl2anc 583 . 2 (𝜑 → (𝐹{⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (dom 𝑅m dom 𝑀) ∧ 𝑔 ∈ (dom 𝑅m dom 𝑀)) ∧ {𝑥 dom 𝑀 ∣ (𝑓𝑥)𝑅(𝑔𝑥)}a.e.𝑀)}𝐺 ↔ ((𝐹 ∈ (dom 𝑅m dom 𝑀) ∧ 𝐺 ∈ (dom 𝑅m dom 𝑀)) ∧ {𝑥 dom 𝑀 ∣ (𝐹𝑥)𝑅(𝐺𝑥)}a.e.𝑀)))
17 brfae.1 . . . 4 (𝜑𝑅 ∈ V)
18 brfae.2 . . . 4 (𝜑𝑀 ran measures)
19 faeval 33865 . . . 4 ((𝑅 ∈ V ∧ 𝑀 ran measures) → (𝑅~ a.e.𝑀) = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (dom 𝑅m dom 𝑀) ∧ 𝑔 ∈ (dom 𝑅m dom 𝑀)) ∧ {𝑥 dom 𝑀 ∣ (𝑓𝑥)𝑅(𝑔𝑥)}a.e.𝑀)})
2017, 18, 19syl2anc 583 . . 3 (𝜑 → (𝑅~ a.e.𝑀) = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (dom 𝑅m dom 𝑀) ∧ 𝑔 ∈ (dom 𝑅m dom 𝑀)) ∧ {𝑥 dom 𝑀 ∣ (𝑓𝑥)𝑅(𝑔𝑥)}a.e.𝑀)})
2120breqd 5159 . 2 (𝜑 → (𝐹(𝑅~ a.e.𝑀)𝐺𝐹{⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (dom 𝑅m dom 𝑀) ∧ 𝑔 ∈ (dom 𝑅m dom 𝑀)) ∧ {𝑥 dom 𝑀 ∣ (𝑓𝑥)𝑅(𝑔𝑥)}a.e.𝑀)}𝐺))
22 brfae.0 . . . . . 6 dom 𝑅 = 𝐷
2322oveq1i 7430 . . . . 5 (dom 𝑅m dom 𝑀) = (𝐷m dom 𝑀)
241, 23eleqtrrdi 2840 . . . 4 (𝜑𝐹 ∈ (dom 𝑅m dom 𝑀))
252, 23eleqtrrdi 2840 . . . 4 (𝜑𝐺 ∈ (dom 𝑅m dom 𝑀))
2624, 25jca 511 . . 3 (𝜑 → (𝐹 ∈ (dom 𝑅m dom 𝑀) ∧ 𝐺 ∈ (dom 𝑅m dom 𝑀)))
2726biantrurd 532 . 2 (𝜑 → ({𝑥 dom 𝑀 ∣ (𝐹𝑥)𝑅(𝐺𝑥)}a.e.𝑀 ↔ ((𝐹 ∈ (dom 𝑅m dom 𝑀) ∧ 𝐺 ∈ (dom 𝑅m dom 𝑀)) ∧ {𝑥 dom 𝑀 ∣ (𝐹𝑥)𝑅(𝐺𝑥)}a.e.𝑀)))
2816, 21, 273bitr4d 311 1 (𝜑 → (𝐹(𝑅~ a.e.𝑀)𝐺 ↔ {𝑥 dom 𝑀 ∣ (𝐹𝑥)𝑅(𝐺𝑥)}a.e.𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1534  wcel 2099  {crab 3429  Vcvv 3471   cuni 4908   class class class wbr 5148  {copab 5210  dom cdm 5678  ran crn 5679  cfv 6548  (class class class)co 7420  m cmap 8844  measurescmeas 33814  a.e.cae 33856  ~ a.e.cfae 33857
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-iota 6500  df-fun 6550  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-fae 33864
This theorem is referenced by: (None)
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