Theorem faeval 33275

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 484	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) → 𝑟 = 𝑅)
2	1	dmeqd 5906	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) → dom 𝑟 = dom 𝑅)
3		simpr 486	. . . . . . . . 9 ⊢ ((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) → 𝑚 = 𝑀)
4	3	dmeqd 5906	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) → dom 𝑚 = dom 𝑀)
5	4	unieqd 4923	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) → ∪ dom 𝑚 = ∪ dom 𝑀)
6	2, 5	oveq12d 7427	. . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) → (dom 𝑟 ↑m ∪ dom 𝑚) = (dom 𝑅 ↑m ∪ dom 𝑀))
7	6	eleq2d 2820	. . . . 5 ⊢ ((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) → (𝑓 ∈ (dom 𝑟 ↑m ∪ dom 𝑚) ↔ 𝑓 ∈ (dom 𝑅 ↑m ∪ dom 𝑀)))
8	6	eleq2d 2820	. . . . 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 5160	. . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) → ((𝑓‘𝑥)𝑟(𝑔‘𝑥) ↔ (𝑓‘𝑥)𝑅(𝑔‘𝑥)))
11	5, 10	rabeqbidv 3450	. . . . 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 5162	. . . 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 5215	. 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 33274	. 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 7442	. . . 4 ⊢ (dom 𝑅 ↑m ∪ dom 𝑀) ∈ V
17	16, 16	xpex 7740	. . 3 ⊢ ((dom 𝑅 ↑m ∪ dom 𝑀) × (dom 𝑅 ↑m ∪ dom 𝑀)) ∈ V
18		opabssxp 5769	. . 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 5323	. 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 7563	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 397   = wceq 1542   ∈ wcel 2107  {crab 3433  Vcvv 3475  ∪ cuni 4909   class class class wbr 5149  {copab 5211   × cxp 5675  dom cdm 5677  ran crn 5678  ‘cfv 6544  (class class class)co 7409   ↑_m cmap 8820  measurescmeas 33224  a.e.cae 33266  ~ a.e.cfae 33267

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-iota 6496  df-fun 6546  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-fae 33274

This theorem is referenced by:  relfae  33276  brfae  33277