Theorem elunirnmbfm 34651

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 34649	. . . . 5 ⊢ MblFnM = (𝑠 ∈ ∪ ran sigAlgebra, 𝑡 ∈ ∪ ran sigAlgebra ↦ {𝑓 ∈ (∪ 𝑡 ↑m ∪ 𝑠) ∣ ∀𝑥 ∈ 𝑡 (◡𝑓 “ 𝑥) ∈ 𝑠})
2	1	mpofun 7536	. . . 4 ⊢ Fun MblFnM
3		elunirn 7249	. . . 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 7445	. . . . . 6 ⊢ (∪ 𝑡 ↑m ∪ 𝑠) ∈ V
6	5	rabex 5308	. . . . 5 ⊢ {𝑓 ∈ (∪ 𝑡 ↑m ∪ 𝑠) ∣ ∀𝑥 ∈ 𝑡 (◡𝑓 “ 𝑥) ∈ 𝑠} ∈ V
7	1, 6	dmmpo 8066	. . . 4 ⊢ dom MblFnM = (∪ ran sigAlgebra × ∪ ran sigAlgebra)
8	7	rexeqi 3321	. . 3 ⊢ (∃𝑎 ∈ dom MblFnM𝐹 ∈ (MblFnM‘𝑎) ↔ ∃𝑎 ∈ (∪ ran sigAlgebra × ∪ ran sigAlgebra)𝐹 ∈ (MblFnM‘𝑎))
9		fveq2 6881	. . . . . 6 ⊢ (𝑎 = ⟨𝑠, 𝑡⟩ → (MblFnM‘𝑎) = (MblFnM‘⟨𝑠, 𝑡⟩))
10		df-ov 7415	. . . . . 6 ⊢ (𝑠MblFnM𝑡) = (MblFnM‘⟨𝑠, 𝑡⟩)
11	9, 10	eqtr4di 2815	. . . . 5 ⊢ (𝑎 = ⟨𝑠, 𝑡⟩ → (MblFnM‘𝑎) = (𝑠MblFnM𝑡))
12	11	eleq2d 2848	. . . 4 ⊢ (𝑎 = ⟨𝑠, 𝑡⟩ → (𝐹 ∈ (MblFnM‘𝑎) ↔ 𝐹 ∈ (𝑠MblFnM𝑡)))
13	12	rexxp 5827	. . 3 ⊢ (∃𝑎 ∈ (∪ ran sigAlgebra × ∪ ran sigAlgebra)𝐹 ∈ (MblFnM‘𝑎) ↔ ∃𝑠 ∈ ∪ ran sigAlgebra∃𝑡 ∈ ∪ ran sigAlgebra𝐹 ∈ (𝑠MblFnM𝑡))
14	4, 8, 13	3bitri 300	. 2 ⊢ (𝐹 ∈ ∪ ran MblFnM ↔ ∃𝑠 ∈ ∪ ran sigAlgebra∃𝑡 ∈ ∪ ran sigAlgebra𝐹 ∈ (𝑠MblFnM𝑡))
15		simpl 487	. . . 4 ⊢ ((𝑠 ∈ ∪ ran sigAlgebra ∧ 𝑡 ∈ ∪ ran sigAlgebra) → 𝑠 ∈ ∪ ran sigAlgebra)
16		simpr 489	. . . 4 ⊢ ((𝑠 ∈ ∪ ran sigAlgebra ∧ 𝑡 ∈ ∪ ran sigAlgebra) → 𝑡 ∈ ∪ ran sigAlgebra)
17	15, 16	ismbfm 34650	. . 3 ⊢ ((𝑠 ∈ ∪ ran sigAlgebra ∧ 𝑡 ∈ ∪ ran sigAlgebra) → (𝐹 ∈ (𝑠MblFnM𝑡) ↔ (𝐹 ∈ (∪ 𝑡 ↑m ∪ 𝑠) ∧ ∀𝑥 ∈ 𝑡 (◡𝐹 “ 𝑥) ∈ 𝑠)))
18	17	2rexbiia 3225	. 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 278	1 ⊢ (𝐹 ∈ ∪ ran MblFnM ↔ ∃𝑠 ∈ ∪ ran sigAlgebra∃𝑡 ∈ ∪ ran sigAlgebra(𝐹 ∈ (∪ 𝑡 ↑m ∪ 𝑠) ∧ ∀𝑥 ∈ 𝑡 (◡𝐹 “ 𝑥) ∈ 𝑠))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 400   = wceq 1569   ∈ wcel 2142  ∀wral 3078  ∃wrex 3088  {crab 3415  ⟨cop 4594  ∪ cuni 4871   × cxp 5658  ^◡ccnv 5659  dom cdm 5660  ran crn 5661   “ cima 5663  Fun wfun 6530  ‘cfv 6536  (class class class)co 7412   ↑_m cmap 8822  sigAlgebracsiga 34507  MblFnMcmbfm 34648

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-nul 5268  ax-pr 5403  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7984  df-2nd 7985  df-mbfm 34649

This theorem is used by:  mbfmfun  34652