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Theorem brfae 31621
 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 486 . . . . . . 7 ((𝑓 = 𝐹𝑔 = 𝐺) → 𝑓 = 𝐹)
43eleq1d 2877 . . . . . 6 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑓 ∈ (dom 𝑅m dom 𝑀) ↔ 𝐹 ∈ (dom 𝑅m dom 𝑀)))
5 simpr 488 . . . . . . 7 ((𝑓 = 𝐹𝑔 = 𝐺) → 𝑔 = 𝐺)
65eleq1d 2877 . . . . . 6 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑔 ∈ (dom 𝑅m dom 𝑀) ↔ 𝐺 ∈ (dom 𝑅m dom 𝑀)))
74, 6anbi12d 633 . . . . 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 6651 . . . . . . . 8 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑓𝑥) = (𝐹𝑥))
95fveq1d 6651 . . . . . . . 8 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑔𝑥) = (𝐺𝑥))
108, 9breq12d 5046 . . . . . . 7 ((𝑓 = 𝐹𝑔 = 𝐺) → ((𝑓𝑥)𝑅(𝑔𝑥) ↔ (𝐹𝑥)𝑅(𝐺𝑥)))
1110rabbidv 3430 . . . . . 6 ((𝑓 = 𝐹𝑔 = 𝐺) → {𝑥 dom 𝑀 ∣ (𝑓𝑥)𝑅(𝑔𝑥)} = {𝑥 dom 𝑀 ∣ (𝐹𝑥)𝑅(𝐺𝑥)})
1211breq1d 5043 . . . . 5 ((𝑓 = 𝐹𝑔 = 𝐺) → ({𝑥 dom 𝑀 ∣ (𝑓𝑥)𝑅(𝑔𝑥)}a.e.𝑀 ↔ {𝑥 dom 𝑀 ∣ (𝐹𝑥)𝑅(𝐺𝑥)}a.e.𝑀))
137, 12anbi12d 633 . . . 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 2801 . . . 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 5389 . . 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 587 . 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 31619 . . . 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 587 . . 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 5044 . 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 7149 . . . . 5 (dom 𝑅m dom 𝑀) = (𝐷m dom 𝑀)
241, 23eleqtrrdi 2904 . . . 4 (𝜑𝐹 ∈ (dom 𝑅m dom 𝑀))
252, 23eleqtrrdi 2904 . . . 4 (𝜑𝐺 ∈ (dom 𝑅m dom 𝑀))
2624, 25jca 515 . . 3 (𝜑 → (𝐹 ∈ (dom 𝑅m dom 𝑀) ∧ 𝐺 ∈ (dom 𝑅m dom 𝑀)))
2726biantrurd 536 . 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 314 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 209   ∧ wa 399   = wceq 1538   ∈ wcel 2112  {crab 3113  Vcvv 3444  ∪ cuni 4803   class class class wbr 5033  {copab 5095  dom cdm 5523  ran crn 5524  ‘cfv 6328  (class class class)co 7139   ↑m cmap 8393  measurescmeas 31568  a.e.cae 31610  ~ a.e.cfae 31611 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 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445 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 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-iota 6287  df-fun 6330  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-fae 31618 This theorem is referenced by: (None)
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