Theorem brfae 33234

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 2818	. . . . . 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 2818	. . . . . 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 6890	. . . . . . . 8 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑓‘𝑥) = (𝐹‘𝑥))
9	5	fveq1d 6890	. . . . . . . 8 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑔‘𝑥) = (𝐺‘𝑥))
10	8, 9	breq12d 5160	. . . . . . 7 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ((𝑓‘𝑥)𝑅(𝑔‘𝑥) ↔ (𝐹‘𝑥)𝑅(𝐺‘𝑥)))
11	10	rabbidv 3440	. . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → {𝑥 ∈ ∪ dom 𝑀 ∣ (𝑓‘𝑥)𝑅(𝑔‘𝑥)} = {𝑥 ∈ ∪ dom 𝑀 ∣ (𝐹‘𝑥)𝑅(𝐺‘𝑥)})
12	11	breq1d 5157	. . . . 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 2732	. . . 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 5533	. . 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 33232	. . . 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 5158	. 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 7415	. . . . 5 ⊢ (dom 𝑅 ↑m ∪ dom 𝑀) = (𝐷 ↑m ∪ dom 𝑀)
24	1, 23	eleqtrrdi 2844	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (dom 𝑅 ↑m ∪ dom 𝑀))
25	2, 23	eleqtrrdi 2844	. . . 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 310	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 1541   ∈ wcel 2106  {crab 3432  Vcvv 3474  ∪ cuni 4907   class class class wbr 5147  {copab 5209  dom cdm 5675  ran crn 5676  ‘cfv 6540  (class class class)co 7405   ↑_m cmap 8816  measurescmeas 33181  a.e.cae 33223  ~ a.e.cfae 33224

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-iota 6492  df-fun 6542  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-fae 33231

This theorem is referenced by: (None)