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Theorem brfae 31158
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 475 . . . . . . 7 ((𝑓 = 𝐹𝑔 = 𝐺) → 𝑓 = 𝐹)
43eleq1d 2850 . . . . . 6 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑓 ∈ (dom 𝑅𝑚 dom 𝑀) ↔ 𝐹 ∈ (dom 𝑅𝑚 dom 𝑀)))
5 simpr 477 . . . . . . 7 ((𝑓 = 𝐹𝑔 = 𝐺) → 𝑔 = 𝐺)
65eleq1d 2850 . . . . . 6 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑔 ∈ (dom 𝑅𝑚 dom 𝑀) ↔ 𝐺 ∈ (dom 𝑅𝑚 dom 𝑀)))
74, 6anbi12d 621 . . . . 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 6501 . . . . . . . 8 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑓𝑥) = (𝐹𝑥))
95fveq1d 6501 . . . . . . . 8 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑔𝑥) = (𝐺𝑥))
108, 9breq12d 4942 . . . . . . 7 ((𝑓 = 𝐹𝑔 = 𝐺) → ((𝑓𝑥)𝑅(𝑔𝑥) ↔ (𝐹𝑥)𝑅(𝐺𝑥)))
1110rabbidv 3403 . . . . . 6 ((𝑓 = 𝐹𝑔 = 𝐺) → {𝑥 dom 𝑀 ∣ (𝑓𝑥)𝑅(𝑔𝑥)} = {𝑥 dom 𝑀 ∣ (𝐹𝑥)𝑅(𝐺𝑥)})
1211breq1d 4939 . . . . 5 ((𝑓 = 𝐹𝑔 = 𝐺) → ({𝑥 dom 𝑀 ∣ (𝑓𝑥)𝑅(𝑔𝑥)}a.e.𝑀 ↔ {𝑥 dom 𝑀 ∣ (𝐹𝑥)𝑅(𝐺𝑥)}a.e.𝑀))
137, 12anbi12d 621 . . . 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 2778 . . . 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 5275 . . 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 576 . 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 31156 . . . 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 576 . . 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 4940 . 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 6986 . . . . 5 (dom 𝑅𝑚 dom 𝑀) = (𝐷𝑚 dom 𝑀)
241, 23syl6eleqr 2877 . . . 4 (𝜑𝐹 ∈ (dom 𝑅𝑚 dom 𝑀))
252, 23syl6eleqr 2877 . . . 4 (𝜑𝐺 ∈ (dom 𝑅𝑚 dom 𝑀))
2624, 25jca 504 . . 3 (𝜑 → (𝐹 ∈ (dom 𝑅𝑚 dom 𝑀) ∧ 𝐺 ∈ (dom 𝑅𝑚 dom 𝑀)))
2726biantrurd 525 . 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 303 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 198  wa 387   = wceq 1507  wcel 2050  {crab 3092  Vcvv 3415   cuni 4712   class class class wbr 4929  {copab 4991  dom cdm 5407  ran crn 5408  cfv 6188  (class class class)co 6976  𝑚 cmap 8206  measurescmeas 31105  a.e.cae 31147  ~ a.e.cfae 31148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3417  df-sbc 3682  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-nul 4179  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-br 4930  df-opab 4992  df-id 5312  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-iota 6152  df-fun 6190  df-fv 6196  df-ov 6979  df-oprab 6980  df-mpo 6981  df-fae 31155
This theorem is referenced by: (None)
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