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Theorem brfae 32887
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 484 . . . . . . 7 ((𝑓 = 𝐹𝑔 = 𝐺) → 𝑓 = 𝐹)
43eleq1d 2823 . . . . . 6 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑓 ∈ (dom 𝑅m dom 𝑀) ↔ 𝐹 ∈ (dom 𝑅m dom 𝑀)))
5 simpr 486 . . . . . . 7 ((𝑓 = 𝐹𝑔 = 𝐺) → 𝑔 = 𝐺)
65eleq1d 2823 . . . . . 6 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑔 ∈ (dom 𝑅m dom 𝑀) ↔ 𝐺 ∈ (dom 𝑅m dom 𝑀)))
74, 6anbi12d 632 . . . . 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 6849 . . . . . . . 8 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑓𝑥) = (𝐹𝑥))
95fveq1d 6849 . . . . . . . 8 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑔𝑥) = (𝐺𝑥))
108, 9breq12d 5123 . . . . . . 7 ((𝑓 = 𝐹𝑔 = 𝐺) → ((𝑓𝑥)𝑅(𝑔𝑥) ↔ (𝐹𝑥)𝑅(𝐺𝑥)))
1110rabbidv 3418 . . . . . 6 ((𝑓 = 𝐹𝑔 = 𝐺) → {𝑥 dom 𝑀 ∣ (𝑓𝑥)𝑅(𝑔𝑥)} = {𝑥 dom 𝑀 ∣ (𝐹𝑥)𝑅(𝐺𝑥)})
1211breq1d 5120 . . . . 5 ((𝑓 = 𝐹𝑔 = 𝐺) → ({𝑥 dom 𝑀 ∣ (𝑓𝑥)𝑅(𝑔𝑥)}a.e.𝑀 ↔ {𝑥 dom 𝑀 ∣ (𝐹𝑥)𝑅(𝐺𝑥)}a.e.𝑀))
137, 12anbi12d 632 . . . 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 2737 . . . 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 5496 . . 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 585 . 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 32885 . . . 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 585 . . 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 5121 . 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 7372 . . . . 5 (dom 𝑅m dom 𝑀) = (𝐷m dom 𝑀)
241, 23eleqtrrdi 2849 . . . 4 (𝜑𝐹 ∈ (dom 𝑅m dom 𝑀))
252, 23eleqtrrdi 2849 . . . 4 (𝜑𝐺 ∈ (dom 𝑅m dom 𝑀))
2624, 25jca 513 . . 3 (𝜑 → (𝐹 ∈ (dom 𝑅m dom 𝑀) ∧ 𝐺 ∈ (dom 𝑅m dom 𝑀)))
2726biantrurd 534 . 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 397   = wceq 1542  wcel 2107  {crab 3410  Vcvv 3448   cuni 4870   class class class wbr 5110  {copab 5172  dom cdm 5638  ran crn 5639  cfv 6501  (class class class)co 7362  m cmap 8772  measurescmeas 32834  a.e.cae 32876  ~ a.e.cfae 32877
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-sbc 3745  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6453  df-fun 6503  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-fae 32884
This theorem is referenced by: (None)
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