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Theorem faeval 31404
Description: Value of the 'almost everywhere' relation for a given relation and measure. (Contributed by Thierry Arnoux, 22-Oct-2017.)
Assertion
Ref Expression
faeval ((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)})
Distinct variable groups:   𝑓,𝑔,𝑥,𝑀   𝑅,𝑓,𝑔,𝑥

Proof of Theorem faeval
Dummy variables 𝑚 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 483 . . . . . . . 8 ((𝑟 = 𝑅𝑚 = 𝑀) → 𝑟 = 𝑅)
21dmeqd 5767 . . . . . . 7 ((𝑟 = 𝑅𝑚 = 𝑀) → dom 𝑟 = dom 𝑅)
3 simpr 485 . . . . . . . . 9 ((𝑟 = 𝑅𝑚 = 𝑀) → 𝑚 = 𝑀)
43dmeqd 5767 . . . . . . . 8 ((𝑟 = 𝑅𝑚 = 𝑀) → dom 𝑚 = dom 𝑀)
54unieqd 4840 . . . . . . 7 ((𝑟 = 𝑅𝑚 = 𝑀) → dom 𝑚 = dom 𝑀)
62, 5oveq12d 7163 . . . . . 6 ((𝑟 = 𝑅𝑚 = 𝑀) → (dom 𝑟m dom 𝑚) = (dom 𝑅m dom 𝑀))
76eleq2d 2895 . . . . 5 ((𝑟 = 𝑅𝑚 = 𝑀) → (𝑓 ∈ (dom 𝑟m dom 𝑚) ↔ 𝑓 ∈ (dom 𝑅m dom 𝑀)))
86eleq2d 2895 . . . . 5 ((𝑟 = 𝑅𝑚 = 𝑀) → (𝑔 ∈ (dom 𝑟m dom 𝑚) ↔ 𝑔 ∈ (dom 𝑅m dom 𝑀)))
97, 8anbi12d 630 . . . 4 ((𝑟 = 𝑅𝑚 = 𝑀) → ((𝑓 ∈ (dom 𝑟m dom 𝑚) ∧ 𝑔 ∈ (dom 𝑟m dom 𝑚)) ↔ (𝑓 ∈ (dom 𝑅m dom 𝑀) ∧ 𝑔 ∈ (dom 𝑅m dom 𝑀))))
101breqd 5068 . . . . . 6 ((𝑟 = 𝑅𝑚 = 𝑀) → ((𝑓𝑥)𝑟(𝑔𝑥) ↔ (𝑓𝑥)𝑅(𝑔𝑥)))
115, 10rabeqbidv 3483 . . . . 5 ((𝑟 = 𝑅𝑚 = 𝑀) → {𝑥 dom 𝑚 ∣ (𝑓𝑥)𝑟(𝑔𝑥)} = {𝑥 dom 𝑀 ∣ (𝑓𝑥)𝑅(𝑔𝑥)})
1211, 3breq12d 5070 . . . 4 ((𝑟 = 𝑅𝑚 = 𝑀) → ({𝑥 dom 𝑚 ∣ (𝑓𝑥)𝑟(𝑔𝑥)}a.e.𝑚 ↔ {𝑥 dom 𝑀 ∣ (𝑓𝑥)𝑅(𝑔𝑥)}a.e.𝑀))
139, 12anbi12d 630 . . 3 ((𝑟 = 𝑅𝑚 = 𝑀) → (((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)))
1413opabbidv 5123 . 2 ((𝑟 = 𝑅𝑚 = 𝑀) → {⟨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)})
15 df-fae 31403 . 2 ~ a.e. = (𝑟 ∈ V, 𝑚 ran measures ↦ {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (dom 𝑟m dom 𝑚) ∧ 𝑔 ∈ (dom 𝑟m dom 𝑚)) ∧ {𝑥 dom 𝑚 ∣ (𝑓𝑥)𝑟(𝑔𝑥)}a.e.𝑚)})
16 ovex 7178 . . . 4 (dom 𝑅m dom 𝑀) ∈ V
1716, 16xpex 7465 . . 3 ((dom 𝑅m dom 𝑀) × (dom 𝑅m dom 𝑀)) ∈ V
18 opabssxp 5636 . . 3 {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (dom 𝑅m dom 𝑀) ∧ 𝑔 ∈ (dom 𝑅m dom 𝑀)) ∧ {𝑥 dom 𝑀 ∣ (𝑓𝑥)𝑅(𝑔𝑥)}a.e.𝑀)} ⊆ ((dom 𝑅m dom 𝑀) × (dom 𝑅m dom 𝑀))
1917, 18ssexi 5217 . 2 {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (dom 𝑅m dom 𝑀) ∧ 𝑔 ∈ (dom 𝑅m dom 𝑀)) ∧ {𝑥 dom 𝑀 ∣ (𝑓𝑥)𝑅(𝑔𝑥)}a.e.𝑀)} ∈ V
2014, 15, 19ovmpoa 7294 1 ((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)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  {crab 3139  Vcvv 3492   cuni 4830   class class class wbr 5057  {copab 5119   × cxp 5546  dom cdm 5548  ran crn 5549  cfv 6348  (class class class)co 7145  m cmap 8395  measurescmeas 31353  a.e.cae 31395  ~ a.e.cfae 31396
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-fae 31403
This theorem is referenced by:  relfae  31405  brfae  31406
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