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Theorem relfae 31580
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 relopab 5673 . 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 31579 . . 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 5630 . 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 261 1 ((𝑅 ∈ V ∧ 𝑀 ran measures) → Rel (𝑅~ a.e.𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wcel 2114  {crab 3134  Vcvv 3469   cuni 4813   class class class wbr 5042  {copab 5104  dom cdm 5532  ran crn 5533  Rel wrel 5537  cfv 6334  (class class class)co 7140  m cmap 8393  measurescmeas 31528  a.e.cae 31570  ~ a.e.cfae 31571
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 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
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 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-iota 6293  df-fun 6336  df-fv 6342  df-ov 7143  df-oprab 7144  df-mpo 7145  df-fae 31578
This theorem is referenced by: (None)
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