Theorem elunirnmbfm 32220

Description: The property of being a measurable function. (Contributed by Thierry Arnoux, 23-Jan-2017.)

Assertion

Ref	Expression
elunirnmbfm	⊢ (𝐹 ∈ ∪ ran MblFnM ↔ ∃𝑠 ∈ ∪ ran sigAlgebra∃𝑡 ∈ ∪ ran sigAlgebra(𝐹 ∈ (∪ 𝑡 ↑m ∪ 𝑠) ∧ ∀𝑥 ∈ 𝑡 (◡𝐹 “ 𝑥) ∈ 𝑠))

Distinct variable group:   𝑡,𝑠,𝐹,𝑥

Proof of Theorem elunirnmbfm

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

Step	Hyp	Ref	Expression
1		df-mbfm 32218	. . . . 5 ⊢ MblFnM = (𝑠 ∈ ∪ ran sigAlgebra, 𝑡 ∈ ∪ ran sigAlgebra ↦ {𝑓 ∈ (∪ 𝑡 ↑m ∪ 𝑠) ∣ ∀𝑥 ∈ 𝑡 (◡𝑓 “ 𝑥) ∈ 𝑠})
2	1	mpofun 7398	. . . 4 ⊢ Fun MblFnM
3		elunirn 7124	. . . 4 ⊢ (Fun MblFnM → (𝐹 ∈ ∪ ran MblFnM ↔ ∃𝑎 ∈ dom MblFnM𝐹 ∈ (MblFnM‘𝑎)))
4	2, 3	ax-mp 5	. . 3 ⊢ (𝐹 ∈ ∪ ran MblFnM ↔ ∃𝑎 ∈ dom MblFnM𝐹 ∈ (MblFnM‘𝑎))
5		ovex 7308	. . . . . 6 ⊢ (∪ 𝑡 ↑m ∪ 𝑠) ∈ V
6	5	rabex 5256	. . . . 5 ⊢ {𝑓 ∈ (∪ 𝑡 ↑m ∪ 𝑠) ∣ ∀𝑥 ∈ 𝑡 (◡𝑓 “ 𝑥) ∈ 𝑠} ∈ V
7	1, 6	dmmpo 7911	. . . 4 ⊢ dom MblFnM = (∪ ran sigAlgebra × ∪ ran sigAlgebra)
8	7	rexeqi 3347	. . 3 ⊢ (∃𝑎 ∈ dom MblFnM𝐹 ∈ (MblFnM‘𝑎) ↔ ∃𝑎 ∈ (∪ ran sigAlgebra × ∪ ran sigAlgebra)𝐹 ∈ (MblFnM‘𝑎))
9		fveq2 6774	. . . . . 6 ⊢ (𝑎 = ⟨𝑠, 𝑡⟩ → (MblFnM‘𝑎) = (MblFnM‘⟨𝑠, 𝑡⟩))
10		df-ov 7278	. . . . . 6 ⊢ (𝑠MblFnM𝑡) = (MblFnM‘⟨𝑠, 𝑡⟩)
11	9, 10	eqtr4di 2796	. . . . 5 ⊢ (𝑎 = ⟨𝑠, 𝑡⟩ → (MblFnM‘𝑎) = (𝑠MblFnM𝑡))
12	11	eleq2d 2824	. . . 4 ⊢ (𝑎 = ⟨𝑠, 𝑡⟩ → (𝐹 ∈ (MblFnM‘𝑎) ↔ 𝐹 ∈ (𝑠MblFnM𝑡)))
13	12	rexxp 5751	. . 3 ⊢ (∃𝑎 ∈ (∪ ran sigAlgebra × ∪ ran sigAlgebra)𝐹 ∈ (MblFnM‘𝑎) ↔ ∃𝑠 ∈ ∪ ran sigAlgebra∃𝑡 ∈ ∪ ran sigAlgebra𝐹 ∈ (𝑠MblFnM𝑡))
14	4, 8, 13	3bitri 297	. 2 ⊢ (𝐹 ∈ ∪ ran MblFnM ↔ ∃𝑠 ∈ ∪ ran sigAlgebra∃𝑡 ∈ ∪ ran sigAlgebra𝐹 ∈ (𝑠MblFnM𝑡))
15		simpl 483	. . . 4 ⊢ ((𝑠 ∈ ∪ ran sigAlgebra ∧ 𝑡 ∈ ∪ ran sigAlgebra) → 𝑠 ∈ ∪ ran sigAlgebra)
16		simpr 485	. . . 4 ⊢ ((𝑠 ∈ ∪ ran sigAlgebra ∧ 𝑡 ∈ ∪ ran sigAlgebra) → 𝑡 ∈ ∪ ran sigAlgebra)
17	15, 16	ismbfm 32219	. . 3 ⊢ ((𝑠 ∈ ∪ ran sigAlgebra ∧ 𝑡 ∈ ∪ ran sigAlgebra) → (𝐹 ∈ (𝑠MblFnM𝑡) ↔ (𝐹 ∈ (∪ 𝑡 ↑m ∪ 𝑠) ∧ ∀𝑥 ∈ 𝑡 (◡𝐹 “ 𝑥) ∈ 𝑠)))
18	17	2rexbiia 3227	. 2 ⊢ (∃𝑠 ∈ ∪ ran sigAlgebra∃𝑡 ∈ ∪ ran sigAlgebra𝐹 ∈ (𝑠MblFnM𝑡) ↔ ∃𝑠 ∈ ∪ ran sigAlgebra∃𝑡 ∈ ∪ ran sigAlgebra(𝐹 ∈ (∪ 𝑡 ↑m ∪ 𝑠) ∧ ∀𝑥 ∈ 𝑡 (◡𝐹 “ 𝑥) ∈ 𝑠))
19	14, 18	bitri 274	1 ⊢ (𝐹 ∈ ∪ ran MblFnM ↔ ∃𝑠 ∈ ∪ ran sigAlgebra∃𝑡 ∈ ∪ ran sigAlgebra(𝐹 ∈ (∪ 𝑡 ↑m ∪ 𝑠) ∧ ∀𝑥 ∈ 𝑡 (◡𝐹 “ 𝑥) ∈ 𝑠))

Syntax hints:   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065  {crab 3068  ⟨cop 4567  ∪ cuni 4839   × cxp 5587  ^◡ccnv 5588  dom cdm 5589  ran crn 5590   “ cima 5592  Fun wfun 6427  ‘cfv 6433  (class class class)co 7275   ↑_m cmap 8615  sigAlgebracsiga 32076  MblFnMcmbfm 32217

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-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-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-mbfm 32218

This theorem is referenced by:  mbfmfun  32221  isanmbfm  32223