Theorem brfae 32216

Description: 'almost everywhere' relation for two functions 𝐹 and 𝐺 with regard to the measure 𝑀. (Contributed by Thierry Arnoux, 22-Oct-2017.)

Hypotheses

Ref	Expression
brfae.0	⊢ dom 𝑅 = 𝐷
brfae.1	⊢ (𝜑 → 𝑅 ∈ V)
brfae.2	⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)
brfae.3	⊢ (𝜑 → 𝐹 ∈ (𝐷 ↑m ∪ dom 𝑀))
brfae.4	⊢ (𝜑 → 𝐺 ∈ (𝐷 ↑m ∪ dom 𝑀))

Assertion

Ref	Expression
brfae	⊢ (𝜑 → (𝐹(𝑅~ a.e.𝑀)𝐺 ↔ {𝑥 ∈ ∪ dom 𝑀 ∣ (𝐹‘𝑥)𝑅(𝐺‘𝑥)}a.e.𝑀))

Distinct variable groups:   𝑥,𝐹   𝑥,𝐺   𝑥,𝑀   𝑥,𝑅

Allowed substitution hints:   𝜑(𝑥)   𝐷(𝑥)

Proof of Theorem brfae

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

Step	Hyp	Ref	Expression
1		brfae.3	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝐷 ↑m ∪ dom 𝑀))
2		brfae.4	. . 3 ⊢ (𝜑 → 𝐺 ∈ (𝐷 ↑m ∪ dom 𝑀))
3		simpl 483	. . . . . . 7 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → 𝑓 = 𝐹)
4	3	eleq1d 2823	. . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑓 ∈ (dom 𝑅 ↑m ∪ dom 𝑀) ↔ 𝐹 ∈ (dom 𝑅 ↑m ∪ dom 𝑀)))
5		simpr 485	. . . . . . 7 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → 𝑔 = 𝐺)
6	5	eleq1d 2823	. . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑔 ∈ (dom 𝑅 ↑m ∪ dom 𝑀) ↔ 𝐺 ∈ (dom 𝑅 ↑m ∪ dom 𝑀)))
7	4, 6	anbi12d 631	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ((𝑓 ∈ (dom 𝑅 ↑m ∪ dom 𝑀) ∧ 𝑔 ∈ (dom 𝑅 ↑m ∪ dom 𝑀)) ↔ (𝐹 ∈ (dom 𝑅 ↑m ∪ dom 𝑀) ∧ 𝐺 ∈ (dom 𝑅 ↑m ∪ dom 𝑀))))
8	3	fveq1d 6776	. . . . . . . 8 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑓‘𝑥) = (𝐹‘𝑥))
9	5	fveq1d 6776	. . . . . . . 8 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑔‘𝑥) = (𝐺‘𝑥))
10	8, 9	breq12d 5087	. . . . . . 7 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ((𝑓‘𝑥)𝑅(𝑔‘𝑥) ↔ (𝐹‘𝑥)𝑅(𝐺‘𝑥)))
11	10	rabbidv 3414	. . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → {𝑥 ∈ ∪ dom 𝑀 ∣ (𝑓‘𝑥)𝑅(𝑔‘𝑥)} = {𝑥 ∈ ∪ dom 𝑀 ∣ (𝐹‘𝑥)𝑅(𝐺‘𝑥)})
12	11	breq1d 5084	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ({𝑥 ∈ ∪ dom 𝑀 ∣ (𝑓‘𝑥)𝑅(𝑔‘𝑥)}a.e.𝑀 ↔ {𝑥 ∈ ∪ dom 𝑀 ∣ (𝐹‘𝑥)𝑅(𝐺‘𝑥)}a.e.𝑀))
13	7, 12	anbi12d 631	. . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 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 ∈ (dom 𝑅 ↑m ∪ dom 𝑀) ∧ 𝐺 ∈ (dom 𝑅 ↑m ∪ dom 𝑀)) ∧ {𝑥 ∈ ∪ dom 𝑀 ∣ (𝐹‘𝑥)𝑅(𝐺‘𝑥)}a.e.𝑀)))
14		eqid 2738	. . . 4 ⊢ {⟨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	13, 14	brabga 5447	. . 3 ⊢ ((𝐹 ∈ (𝐷 ↑m ∪ dom 𝑀) ∧ 𝐺 ∈ (𝐷 ↑m ∪ dom 𝑀)) → (𝐹{⟨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)}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	1, 2, 15	syl2anc 584	. 2 ⊢ (𝜑 → (𝐹{⟨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)}𝐺 ↔ ((𝐹 ∈ (dom 𝑅 ↑m ∪ dom 𝑀) ∧ 𝐺 ∈ (dom 𝑅 ↑m ∪ dom 𝑀)) ∧ {𝑥 ∈ ∪ dom 𝑀 ∣ (𝐹‘𝑥)𝑅(𝐺‘𝑥)}a.e.𝑀)))
17		brfae.1	. . . 4 ⊢ (𝜑 → 𝑅 ∈ V)
18		brfae.2	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)
19		faeval 32214	. . . 4 ⊢ ((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)})
20	17, 18, 19	syl2anc 584	. . 3 ⊢ (𝜑 → (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)})
21	20	breqd 5085	. 2 ⊢ (𝜑 → (𝐹(𝑅~ a.e.𝑀)𝐺 ↔ 𝐹{⟨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)}𝐺))
22		brfae.0	. . . . . 6 ⊢ dom 𝑅 = 𝐷
23	22	oveq1i 7285	. . . . 5 ⊢ (dom 𝑅 ↑m ∪ dom 𝑀) = (𝐷 ↑m ∪ dom 𝑀)
24	1, 23	eleqtrrdi 2850	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (dom 𝑅 ↑m ∪ dom 𝑀))
25	2, 23	eleqtrrdi 2850	. . . 4 ⊢ (𝜑 → 𝐺 ∈ (dom 𝑅 ↑m ∪ dom 𝑀))
26	24, 25	jca 512	. . 3 ⊢ (𝜑 → (𝐹 ∈ (dom 𝑅 ↑m ∪ dom 𝑀) ∧ 𝐺 ∈ (dom 𝑅 ↑m ∪ dom 𝑀)))
27	26	biantrurd 533	. 2 ⊢ (𝜑 → ({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)))
28	16, 21, 27	3bitr4d 311	1 ⊢ (𝜑 → (𝐹(𝑅~ a.e.𝑀)𝐺 ↔ {𝑥 ∈ ∪ dom 𝑀 ∣ (𝐹‘𝑥)𝑅(𝐺‘𝑥)}a.e.𝑀))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  {crab 3068  Vcvv 3432  ∪ cuni 4839   class class class wbr 5074  {copab 5136  dom cdm 5589  ran crn 5590  ‘cfv 6433  (class class class)co 7275   ↑_m cmap 8615  measurescmeas 32163  a.e.cae 32205  ~ a.e.cfae 32206

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-iota 6391  df-fun 6435  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-fae 32213

This theorem is referenced by: (None)