Theorem faeval 31500

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 𝑅 ↑m ∪ dom 𝑀) ∧ 𝑔 ∈ (dom 𝑅 ↑m ∪ dom 𝑀)) ∧ {𝑥 ∈ ∪ dom 𝑀 ∣ (𝑓‘𝑥)𝑅(𝑔‘𝑥)}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.

Step	Hyp	Ref	Expression
1		simpl 485	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) → 𝑟 = 𝑅)
2	1	dmeqd 5768	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) → dom 𝑟 = dom 𝑅)
3		simpr 487	. . . . . . . . 9 ⊢ ((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) → 𝑚 = 𝑀)
4	3	dmeqd 5768	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) → dom 𝑚 = dom 𝑀)
5	4	unieqd 4841	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) → ∪ dom 𝑚 = ∪ dom 𝑀)
6	2, 5	oveq12d 7168	. . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) → (dom 𝑟 ↑m ∪ dom 𝑚) = (dom 𝑅 ↑m ∪ dom 𝑀))
7	6	eleq2d 2898	. . . . 5 ⊢ ((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) → (𝑓 ∈ (dom 𝑟 ↑m ∪ dom 𝑚) ↔ 𝑓 ∈ (dom 𝑅 ↑m ∪ dom 𝑀)))
8	6	eleq2d 2898	. . . . 5 ⊢ ((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) → (𝑔 ∈ (dom 𝑟 ↑m ∪ dom 𝑚) ↔ 𝑔 ∈ (dom 𝑅 ↑m ∪ dom 𝑀)))
9	7, 8	anbi12d 632	. . . 4 ⊢ ((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) → ((𝑓 ∈ (dom 𝑟 ↑m ∪ dom 𝑚) ∧ 𝑔 ∈ (dom 𝑟 ↑m ∪ dom 𝑚)) ↔ (𝑓 ∈ (dom 𝑅 ↑m ∪ dom 𝑀) ∧ 𝑔 ∈ (dom 𝑅 ↑m ∪ dom 𝑀))))
10	1	breqd 5069	. . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) → ((𝑓‘𝑥)𝑟(𝑔‘𝑥) ↔ (𝑓‘𝑥)𝑅(𝑔‘𝑥)))
11	5, 10	rabeqbidv 3485	. . . . 5 ⊢ ((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) → {𝑥 ∈ ∪ dom 𝑚 ∣ (𝑓‘𝑥)𝑟(𝑔‘𝑥)} = {𝑥 ∈ ∪ dom 𝑀 ∣ (𝑓‘𝑥)𝑅(𝑔‘𝑥)})
12	11, 3	breq12d 5071	. . . 4 ⊢ ((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) → ({𝑥 ∈ ∪ dom 𝑚 ∣ (𝑓‘𝑥)𝑟(𝑔‘𝑥)}a.e.𝑚 ↔ {𝑥 ∈ ∪ dom 𝑀 ∣ (𝑓‘𝑥)𝑅(𝑔‘𝑥)}a.e.𝑀))
13	9, 12	anbi12d 632	. . 3 ⊢ ((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) → (((𝑓 ∈ (dom 𝑟 ↑m ∪ dom 𝑚) ∧ 𝑔 ∈ (dom 𝑟 ↑m ∪ dom 𝑚)) ∧ {𝑥 ∈ ∪ dom 𝑚 ∣ (𝑓‘𝑥)𝑟(𝑔‘𝑥)}a.e.𝑚) ↔ ((𝑓 ∈ (dom 𝑅 ↑m ∪ dom 𝑀) ∧ 𝑔 ∈ (dom 𝑅 ↑m ∪ dom 𝑀)) ∧ {𝑥 ∈ ∪ dom 𝑀 ∣ (𝑓‘𝑥)𝑅(𝑔‘𝑥)}a.e.𝑀)))
14	13	opabbidv 5124	. 2 ⊢ ((𝑟 = 𝑅 ∧ 𝑚 = 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)} = {⟨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)})
15		df-fae 31499	. 2 ⊢ ~ a.e. = (𝑟 ∈ V, 𝑚 ∈ ∪ ran measures ↦ {⟨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)})
16		ovex 7183	. . . 4 ⊢ (dom 𝑅 ↑m ∪ dom 𝑀) ∈ V
17	16, 16	xpex 7470	. . 3 ⊢ ((dom 𝑅 ↑m ∪ dom 𝑀) × (dom 𝑅 ↑m ∪ dom 𝑀)) ∈ V
18		opabssxp 5637	. . 3 ⊢ {⟨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)} ⊆ ((dom 𝑅 ↑m ∪ dom 𝑀) × (dom 𝑅 ↑m ∪ dom 𝑀))
19	17, 18	ssexi 5218	. 2 ⊢ {⟨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)} ∈ V
20	14, 15, 19	ovmpoa 7299	1 ⊢ ((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)})

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1533   ∈ wcel 2110  {crab 3142  Vcvv 3494  ∪ cuni 4831   class class class wbr 5058  {copab 5120   × cxp 5547  dom cdm 5549  ran crn 5550  ‘cfv 6349  (class class class)co 7150   ↑_m cmap 8400  measurescmeas 31449  a.e.cae 31491  ~ a.e.cfae 31492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-opab 5121  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-iota 6308  df-fun 6351  df-fv 6357  df-ov 7153  df-oprab 7154  df-mpo 7155  df-fae 31499

This theorem is referenced by:  relfae  31501  brfae  31502