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Theorem brfae 30612
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 474 . . . . . . 7 ((𝑓 = 𝐹𝑔 = 𝐺) → 𝑓 = 𝐹)
43eleq1d 2816 . . . . . 6 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑓 ∈ (dom 𝑅𝑚 dom 𝑀) ↔ 𝐹 ∈ (dom 𝑅𝑚 dom 𝑀)))
5 simpr 479 . . . . . . 7 ((𝑓 = 𝐹𝑔 = 𝐺) → 𝑔 = 𝐺)
65eleq1d 2816 . . . . . 6 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑔 ∈ (dom 𝑅𝑚 dom 𝑀) ↔ 𝐺 ∈ (dom 𝑅𝑚 dom 𝑀)))
74, 6anbi12d 749 . . . . 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 6346 . . . . . . . 8 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑓𝑥) = (𝐹𝑥))
95fveq1d 6346 . . . . . . . 8 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑔𝑥) = (𝐺𝑥))
108, 9breq12d 4809 . . . . . . 7 ((𝑓 = 𝐹𝑔 = 𝐺) → ((𝑓𝑥)𝑅(𝑔𝑥) ↔ (𝐹𝑥)𝑅(𝐺𝑥)))
1110rabbidv 3321 . . . . . 6 ((𝑓 = 𝐹𝑔 = 𝐺) → {𝑥 dom 𝑀 ∣ (𝑓𝑥)𝑅(𝑔𝑥)} = {𝑥 dom 𝑀 ∣ (𝐹𝑥)𝑅(𝐺𝑥)})
1211breq1d 4806 . . . . 5 ((𝑓 = 𝐹𝑔 = 𝐺) → ({𝑥 dom 𝑀 ∣ (𝑓𝑥)𝑅(𝑔𝑥)}a.e.𝑀 ↔ {𝑥 dom 𝑀 ∣ (𝐹𝑥)𝑅(𝐺𝑥)}a.e.𝑀))
137, 12anbi12d 749 . . . 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 2752 . . . 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 5131 . . 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 696 . 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 30610 . . . 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 696 . . 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 4807 . 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 6815 . . . . 5 (dom 𝑅𝑚 dom 𝑀) = (𝐷𝑚 dom 𝑀)
241, 23syl6eleqr 2842 . . . 4 (𝜑𝐹 ∈ (dom 𝑅𝑚 dom 𝑀))
252, 23syl6eleqr 2842 . . . 4 (𝜑𝐺 ∈ (dom 𝑅𝑚 dom 𝑀))
2624, 25jca 555 . . 3 (𝜑 → (𝐹 ∈ (dom 𝑅𝑚 dom 𝑀) ∧ 𝐺 ∈ (dom 𝑅𝑚 dom 𝑀)))
2726biantrurd 530 . 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 300 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 196  wa 383   = wceq 1624  wcel 2131  {crab 3046  Vcvv 3332   cuni 4580   class class class wbr 4796  {copab 4856  dom cdm 5258  ran crn 5259  cfv 6041  (class class class)co 6805  𝑚 cmap 8015  measurescmeas 30559  a.e.cae 30601  ~ a.e.cfae 30602
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-sbc 3569  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-br 4797  df-opab 4857  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-iota 6004  df-fun 6043  df-fv 6049  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-fae 30609
This theorem is referenced by: (None)
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