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Theorem faeval 34226
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 482 . . . . . . . 8 ((𝑟 = 𝑅𝑚 = 𝑀) → 𝑟 = 𝑅)
21dmeqd 5918 . . . . . . 7 ((𝑟 = 𝑅𝑚 = 𝑀) → dom 𝑟 = dom 𝑅)
3 simpr 484 . . . . . . . . 9 ((𝑟 = 𝑅𝑚 = 𝑀) → 𝑚 = 𝑀)
43dmeqd 5918 . . . . . . . 8 ((𝑟 = 𝑅𝑚 = 𝑀) → dom 𝑚 = dom 𝑀)
54unieqd 4924 . . . . . . 7 ((𝑟 = 𝑅𝑚 = 𝑀) → dom 𝑚 = dom 𝑀)
62, 5oveq12d 7448 . . . . . 6 ((𝑟 = 𝑅𝑚 = 𝑀) → (dom 𝑟m dom 𝑚) = (dom 𝑅m dom 𝑀))
76eleq2d 2824 . . . . 5 ((𝑟 = 𝑅𝑚 = 𝑀) → (𝑓 ∈ (dom 𝑟m dom 𝑚) ↔ 𝑓 ∈ (dom 𝑅m dom 𝑀)))
86eleq2d 2824 . . . . 5 ((𝑟 = 𝑅𝑚 = 𝑀) → (𝑔 ∈ (dom 𝑟m dom 𝑚) ↔ 𝑔 ∈ (dom 𝑅m dom 𝑀)))
97, 8anbi12d 632 . . . 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 5158 . . . . . 6 ((𝑟 = 𝑅𝑚 = 𝑀) → ((𝑓𝑥)𝑟(𝑔𝑥) ↔ (𝑓𝑥)𝑅(𝑔𝑥)))
115, 10rabeqbidv 3451 . . . . 5 ((𝑟 = 𝑅𝑚 = 𝑀) → {𝑥 dom 𝑚 ∣ (𝑓𝑥)𝑟(𝑔𝑥)} = {𝑥 dom 𝑀 ∣ (𝑓𝑥)𝑅(𝑔𝑥)})
1211, 3breq12d 5160 . . . 4 ((𝑟 = 𝑅𝑚 = 𝑀) → ({𝑥 dom 𝑚 ∣ (𝑓𝑥)𝑟(𝑔𝑥)}a.e.𝑚 ↔ {𝑥 dom 𝑀 ∣ (𝑓𝑥)𝑅(𝑔𝑥)}a.e.𝑀))
139, 12anbi12d 632 . . 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 5213 . 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 34225 . 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 7463 . . . 4 (dom 𝑅m dom 𝑀) ∈ V
1716, 16xpex 7771 . . 3 ((dom 𝑅m dom 𝑀) × (dom 𝑅m dom 𝑀)) ∈ V
18 opabssxp 5780 . . 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 5327 . 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 7587 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 395   = wceq 1536  wcel 2105  {crab 3432  Vcvv 3477   cuni 4911   class class class wbr 5147  {copab 5209   × cxp 5686  dom cdm 5688  ran crn 5689  cfv 6562  (class class class)co 7430  m cmap 8864  measurescmeas 34175  a.e.cae 34217  ~ a.e.cfae 34218
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-iota 6515  df-fun 6564  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-fae 34225
This theorem is referenced by:  relfae  34227  brfae  34228
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