Theorem faeval 34403

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 5854	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) → dom 𝑟 = dom 𝑅)
3		simpr 484	. . . . . . . . 9 ⊢ ((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) → 𝑚 = 𝑀)
4	3	dmeqd 5854	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) → dom 𝑚 = dom 𝑀)
5	4	unieqd 4876	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) → ∪ dom 𝑚 = ∪ dom 𝑀)
6	2, 5	oveq12d 7376	. . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) → (dom 𝑟 ↑m ∪ dom 𝑚) = (dom 𝑅 ↑m ∪ dom 𝑀))
7	6	eleq2d 2822	. . . . 5 ⊢ ((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) → (𝑓 ∈ (dom 𝑟 ↑m ∪ dom 𝑚) ↔ 𝑓 ∈ (dom 𝑅 ↑m ∪ dom 𝑀)))
8	6	eleq2d 2822	. . . . 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 5109	. . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) → ((𝑓‘𝑥)𝑟(𝑔‘𝑥) ↔ (𝑓‘𝑥)𝑅(𝑔‘𝑥)))
11	5, 10	rabeqbidv 3417	. . . . 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 5111	. . . 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 5164	. 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 34402	. 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 7391	. . . 4 ⊢ (dom 𝑅 ↑m ∪ dom 𝑀) ∈ V
17	16, 16	xpex 7698	. . 3 ⊢ ((dom 𝑅 ↑m ∪ dom 𝑀) × (dom 𝑅 ↑m ∪ dom 𝑀)) ∈ V
18		opabssxp 5716	. . 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 5267	. 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 7513	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 1541   ∈ wcel 2113  {crab 3399  Vcvv 3440  ∪ cuni 4863   class class class wbr 5098  {copab 5160   × cxp 5622  dom cdm 5624  ran crn 5625  ‘cfv 6492  (class class class)co 7358   ↑_m cmap 8763  measurescmeas 34352  a.e.cae 34394  ~ a.e.cfae 34395

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-iota 6448  df-fun 6494  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-fae 34402

This theorem is referenced by:  relfae  34404  brfae  34405