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Theorem relfae 34248
Description: The 'almost everywhere' builder for functions produces relations. (Contributed by Thierry Arnoux, 22-Oct-2017.)
Assertion
Ref Expression
relfae ((𝑅 ∈ V ∧ 𝑀 ran measures) → Rel (𝑅~ a.e.𝑀))

Proof of Theorem relfae
Dummy variables 𝑓 𝑔 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relopabv 5831 . 2 Rel {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (dom 𝑅m dom 𝑀) ∧ 𝑔 ∈ (dom 𝑅m dom 𝑀)) ∧ {𝑥 dom 𝑀 ∣ (𝑓𝑥)𝑅(𝑔𝑥)}a.e.𝑀)}
2 faeval 34247 . . 3 ((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)})
32releqd 5788 . 2 ((𝑅 ∈ V ∧ 𝑀 ran measures) → (Rel (𝑅~ a.e.𝑀) ↔ Rel {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (dom 𝑅m dom 𝑀) ∧ 𝑔 ∈ (dom 𝑅m dom 𝑀)) ∧ {𝑥 dom 𝑀 ∣ (𝑓𝑥)𝑅(𝑔𝑥)}a.e.𝑀)}))
41, 3mpbiri 258 1 ((𝑅 ∈ V ∧ 𝑀 ran measures) → Rel (𝑅~ a.e.𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2108  {crab 3436  Vcvv 3480   cuni 4907   class class class wbr 5143  {copab 5205  dom cdm 5685  ran crn 5686  Rel wrel 5690  cfv 6561  (class class class)co 7431  m cmap 8866  measurescmeas 34196  a.e.cae 34238  ~ a.e.cfae 34239
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-fae 34246
This theorem is referenced by: (None)
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