Theorem relfae 34431

Description: The 'almost everywhere' builder for functions produces relations. (Contributed by Thierry Arnoux, 22-Oct-2017.)

Assertion

Ref	Expression
relfae	⊢ ((𝑅 ∈ V ∧ 𝑀 ∈ ∪ ran measures) → Rel (𝑅~ a.e.𝑀))

Proof of Theorem relfae

Dummy variables 𝑓 𝑔 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relopabv 5764	. 2 ⊢ Rel {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (dom 𝑅 ↑m ∪ dom 𝑀) ∧ 𝑔 ∈ (dom 𝑅 ↑m ∪ dom 𝑀)) ∧ {𝑥 ∈ ∪ dom 𝑀 ∣ (𝑓‘𝑥)𝑅(𝑔‘𝑥)}a.e.𝑀)}
2		faeval 34430	. . 3 ⊢ ((𝑅 ∈ V ∧ 𝑀 ∈ ∪ ran measures) → (𝑅~ a.e.𝑀) = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (dom 𝑅 ↑m ∪ dom 𝑀) ∧ 𝑔 ∈ (dom 𝑅 ↑m ∪ dom 𝑀)) ∧ {𝑥 ∈ ∪ dom 𝑀 ∣ (𝑓‘𝑥)𝑅(𝑔‘𝑥)}a.e.𝑀)})
3	2	releqd 5722	. 2 ⊢ ((𝑅 ∈ V ∧ 𝑀 ∈ ∪ ran measures) → (Rel (𝑅~ a.e.𝑀) ↔ Rel {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (dom 𝑅 ↑m ∪ dom 𝑀) ∧ 𝑔 ∈ (dom 𝑅 ↑m ∪ dom 𝑀)) ∧ {𝑥 ∈ ∪ dom 𝑀 ∣ (𝑓‘𝑥)𝑅(𝑔‘𝑥)}a.e.𝑀)}))
4	1, 3	mpbiri 259	1 ⊢ ((𝑅 ∈ V ∧ 𝑀 ∈ ∪ ran measures) → Rel (𝑅~ a.e.𝑀))

Syntax hints:   → wi 4   ∧ wa 396   ∈ wcel 2119  {crab 3391  Vcvv 3431  ∪ cuni 4838   class class class wbr 5072  {copab 5134  dom cdm 5618  ran crn 5619  Rel wrel 5623  ‘cfv 6485  (class class class)co 7356   ↑_m cmap 8763  measurescmeas 34379  a.e.cae 34421  ~ a.e.cfae 34422

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-sbc 3724  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-iota 6441  df-fun 6487  df-fv 6493  df-ov 7359  df-oprab 7360  df-mpo 7361  df-fae 34429

This theorem is referenced by: (None)