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Theorem faeval 31615
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 486 . . . . . . . 8 ((𝑟 = 𝑅𝑚 = 𝑀) → 𝑟 = 𝑅)
21dmeqd 5738 . . . . . . 7 ((𝑟 = 𝑅𝑚 = 𝑀) → dom 𝑟 = dom 𝑅)
3 simpr 488 . . . . . . . . 9 ((𝑟 = 𝑅𝑚 = 𝑀) → 𝑚 = 𝑀)
43dmeqd 5738 . . . . . . . 8 ((𝑟 = 𝑅𝑚 = 𝑀) → dom 𝑚 = dom 𝑀)
54unieqd 4814 . . . . . . 7 ((𝑟 = 𝑅𝑚 = 𝑀) → dom 𝑚 = dom 𝑀)
62, 5oveq12d 7153 . . . . . 6 ((𝑟 = 𝑅𝑚 = 𝑀) → (dom 𝑟m dom 𝑚) = (dom 𝑅m dom 𝑀))
76eleq2d 2875 . . . . 5 ((𝑟 = 𝑅𝑚 = 𝑀) → (𝑓 ∈ (dom 𝑟m dom 𝑚) ↔ 𝑓 ∈ (dom 𝑅m dom 𝑀)))
86eleq2d 2875 . . . . 5 ((𝑟 = 𝑅𝑚 = 𝑀) → (𝑔 ∈ (dom 𝑟m dom 𝑚) ↔ 𝑔 ∈ (dom 𝑅m dom 𝑀)))
97, 8anbi12d 633 . . . 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 5041 . . . . . 6 ((𝑟 = 𝑅𝑚 = 𝑀) → ((𝑓𝑥)𝑟(𝑔𝑥) ↔ (𝑓𝑥)𝑅(𝑔𝑥)))
115, 10rabeqbidv 3433 . . . . 5 ((𝑟 = 𝑅𝑚 = 𝑀) → {𝑥 dom 𝑚 ∣ (𝑓𝑥)𝑟(𝑔𝑥)} = {𝑥 dom 𝑀 ∣ (𝑓𝑥)𝑅(𝑔𝑥)})
1211, 3breq12d 5043 . . . 4 ((𝑟 = 𝑅𝑚 = 𝑀) → ({𝑥 dom 𝑚 ∣ (𝑓𝑥)𝑟(𝑔𝑥)}a.e.𝑚 ↔ {𝑥 dom 𝑀 ∣ (𝑓𝑥)𝑅(𝑔𝑥)}a.e.𝑀))
139, 12anbi12d 633 . . 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 5096 . 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 31614 . 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 7168 . . . 4 (dom 𝑅m dom 𝑀) ∈ V
1716, 16xpex 7456 . . 3 ((dom 𝑅m dom 𝑀) × (dom 𝑅m dom 𝑀)) ∈ V
18 opabssxp 5607 . . 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 5190 . 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 7284 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 399   = wceq 1538  wcel 2111  {crab 3110  Vcvv 3441   cuni 4800   class class class wbr 5030  {copab 5092   × cxp 5517  dom cdm 5519  ran crn 5520  cfv 6324  (class class class)co 7135  m cmap 8389  measurescmeas 31564  a.e.cae 31606  ~ a.e.cfae 31607
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 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441
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 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-iota 6283  df-fun 6326  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-fae 31614
This theorem is referenced by:  relfae  31616  brfae  31617
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