Theorem faeval 32114

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 482	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) → 𝑟 = 𝑅)
2	1	dmeqd 5803	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) → dom 𝑟 = dom 𝑅)
3		simpr 484	. . . . . . . . 9 ⊢ ((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) → 𝑚 = 𝑀)
4	3	dmeqd 5803	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) → dom 𝑚 = dom 𝑀)
5	4	unieqd 4850	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) → ∪ dom 𝑚 = ∪ dom 𝑀)
6	2, 5	oveq12d 7273	. . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) → (dom 𝑟 ↑m ∪ dom 𝑚) = (dom 𝑅 ↑m ∪ dom 𝑀))
7	6	eleq2d 2824	. . . . 5 ⊢ ((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) → (𝑓 ∈ (dom 𝑟 ↑m ∪ dom 𝑚) ↔ 𝑓 ∈ (dom 𝑅 ↑m ∪ dom 𝑀)))
8	6	eleq2d 2824	. . . . 5 ⊢ ((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) → (𝑔 ∈ (dom 𝑟 ↑m ∪ dom 𝑚) ↔ 𝑔 ∈ (dom 𝑅 ↑m ∪ dom 𝑀)))
9	7, 8	anbi12d 630	. . . 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 5081	. . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) → ((𝑓‘𝑥)𝑟(𝑔‘𝑥) ↔ (𝑓‘𝑥)𝑅(𝑔‘𝑥)))
11	5, 10	rabeqbidv 3410	. . . . 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 5083	. . . 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 630	. . 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 5136	. 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 32113	. 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 7288	. . . 4 ⊢ (dom 𝑅 ↑m ∪ dom 𝑀) ∈ V
17	16, 16	xpex 7581	. . 3 ⊢ ((dom 𝑅 ↑m ∪ dom 𝑀) × (dom 𝑅 ↑m ∪ dom 𝑀)) ∈ V
18		opabssxp 5669	. . 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 5241	. 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 7406	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 395   = wceq 1539   ∈ wcel 2108  {crab 3067  Vcvv 3422  ∪ cuni 4836   class class class wbr 5070  {copab 5132   × cxp 5578  dom cdm 5580  ran crn 5581  ‘cfv 6418  (class class class)co 7255   ↑_m cmap 8573  measurescmeas 32063  a.e.cae 32105  ~ a.e.cfae 32106

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-iota 6376  df-fun 6420  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-fae 32113

This theorem is referenced by:  relfae  32115  brfae  32116