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Theorem faeval 33801
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 5902 . . . . . . 7 ((𝑟 = 𝑅𝑚 = 𝑀) → dom 𝑟 = dom 𝑅)
3 simpr 484 . . . . . . . . 9 ((𝑟 = 𝑅𝑚 = 𝑀) → 𝑚 = 𝑀)
43dmeqd 5902 . . . . . . . 8 ((𝑟 = 𝑅𝑚 = 𝑀) → dom 𝑚 = dom 𝑀)
54unieqd 4916 . . . . . . 7 ((𝑟 = 𝑅𝑚 = 𝑀) → dom 𝑚 = dom 𝑀)
62, 5oveq12d 7432 . . . . . 6 ((𝑟 = 𝑅𝑚 = 𝑀) → (dom 𝑟m dom 𝑚) = (dom 𝑅m dom 𝑀))
76eleq2d 2814 . . . . 5 ((𝑟 = 𝑅𝑚 = 𝑀) → (𝑓 ∈ (dom 𝑟m dom 𝑚) ↔ 𝑓 ∈ (dom 𝑅m dom 𝑀)))
86eleq2d 2814 . . . . 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 5153 . . . . . 6 ((𝑟 = 𝑅𝑚 = 𝑀) → ((𝑓𝑥)𝑟(𝑔𝑥) ↔ (𝑓𝑥)𝑅(𝑔𝑥)))
115, 10rabeqbidv 3444 . . . . 5 ((𝑟 = 𝑅𝑚 = 𝑀) → {𝑥 dom 𝑚 ∣ (𝑓𝑥)𝑟(𝑔𝑥)} = {𝑥 dom 𝑀 ∣ (𝑓𝑥)𝑅(𝑔𝑥)})
1211, 3breq12d 5155 . . . 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 5208 . 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 33800 . 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 7447 . . . 4 (dom 𝑅m dom 𝑀) ∈ V
1716, 16xpex 7749 . . 3 ((dom 𝑅m dom 𝑀) × (dom 𝑅m dom 𝑀)) ∈ V
18 opabssxp 5764 . . 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 5316 . 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 7570 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 1534  wcel 2099  {crab 3427  Vcvv 3469   cuni 4903   class class class wbr 5142  {copab 5204   × cxp 5670  dom cdm 5672  ran crn 5673  cfv 6542  (class class class)co 7414  m cmap 8836  measurescmeas 33750  a.e.cae 33792  ~ a.e.cfae 33793
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-sbc 3775  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-iota 6494  df-fun 6544  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-fae 33800
This theorem is referenced by:  relfae  33802  brfae  33803
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