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Theorem faeval 29636
 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 472 . . . . . . . 8 ((𝑟 = 𝑅𝑚 = 𝑀) → 𝑟 = 𝑅)
21dmeqd 5248 . . . . . . 7 ((𝑟 = 𝑅𝑚 = 𝑀) → dom 𝑟 = dom 𝑅)
3 simpr 476 . . . . . . . . 9 ((𝑟 = 𝑅𝑚 = 𝑀) → 𝑚 = 𝑀)
43dmeqd 5248 . . . . . . . 8 ((𝑟 = 𝑅𝑚 = 𝑀) → dom 𝑚 = dom 𝑀)
54unieqd 4382 . . . . . . 7 ((𝑟 = 𝑅𝑚 = 𝑀) → dom 𝑚 = dom 𝑀)
62, 5oveq12d 6567 . . . . . 6 ((𝑟 = 𝑅𝑚 = 𝑀) → (dom 𝑟𝑚 dom 𝑚) = (dom 𝑅𝑚 dom 𝑀))
76eleq2d 2673 . . . . 5 ((𝑟 = 𝑅𝑚 = 𝑀) → (𝑓 ∈ (dom 𝑟𝑚 dom 𝑚) ↔ 𝑓 ∈ (dom 𝑅𝑚 dom 𝑀)))
86eleq2d 2673 . . . . 5 ((𝑟 = 𝑅𝑚 = 𝑀) → (𝑔 ∈ (dom 𝑟𝑚 dom 𝑚) ↔ 𝑔 ∈ (dom 𝑅𝑚 dom 𝑀)))
97, 8anbi12d 743 . . . 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 4594 . . . . . 6 ((𝑟 = 𝑅𝑚 = 𝑀) → ((𝑓𝑥)𝑟(𝑔𝑥) ↔ (𝑓𝑥)𝑅(𝑔𝑥)))
115, 10rabeqbidv 3168 . . . . 5 ((𝑟 = 𝑅𝑚 = 𝑀) → {𝑥 dom 𝑚 ∣ (𝑓𝑥)𝑟(𝑔𝑥)} = {𝑥 dom 𝑀 ∣ (𝑓𝑥)𝑅(𝑔𝑥)})
1211, 3breq12d 4596 . . . 4 ((𝑟 = 𝑅𝑚 = 𝑀) → ({𝑥 dom 𝑚 ∣ (𝑓𝑥)𝑟(𝑔𝑥)}a.e.𝑚 ↔ {𝑥 dom 𝑀 ∣ (𝑓𝑥)𝑅(𝑔𝑥)}a.e.𝑀))
139, 12anbi12d 743 . . 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 4648 . 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 29635 . 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 6577 . . . 4 (dom 𝑅𝑚 dom 𝑀) ∈ V
1716, 16xpex 6860 . . 3 ((dom 𝑅𝑚 dom 𝑀) × (dom 𝑅𝑚 dom 𝑀)) ∈ V
18 opabssxp 5116 . . 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 4731 . 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, 19ovmpt2a 6689 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 383   = wceq 1475   ∈ wcel 1977  {crab 2900  Vcvv 3173  ∪ cuni 4372   class class class wbr 4583  {copab 4642   × cxp 5036  dom cdm 5038  ran crn 5039  ‘cfv 5804  (class class class)co 6549   ↑𝑚 cmap 7744  measurescmeas 29585  a.e.cae 29627  ~ a.e.cfae 29628 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-fae 29635 This theorem is referenced by:  relfae  29637  brfae  29638
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