Theorem elunirnmbfm 34765

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 34763	. . . . 5 ⊢ MblFnM = (𝑠 ∈ ∪ ran sigAlgebra, 𝑡 ∈ ∪ ran sigAlgebra ↦ {𝑓 ∈ (∪ 𝑡 ↑m ∪ 𝑠) ∣ ∀𝑥 ∈ 𝑡 (◡𝑓 “ 𝑥) ∈ 𝑠})
2	1	mpofun 7540	. . . 4 ⊢ Fun MblFnM
3		elunirn 7251	. . . 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 7449	. . . . . 6 ⊢ (∪ 𝑡 ↑m ∪ 𝑠) ∈ V
6	5	rabex 5307	. . . . 5 ⊢ {𝑓 ∈ (∪ 𝑡 ↑m ∪ 𝑠) ∣ ∀𝑥 ∈ 𝑡 (◡𝑓 “ 𝑥) ∈ 𝑠} ∈ V
7	1, 6	dmmpo 8071	. . . 4 ⊢ dom MblFnM = (∪ ran sigAlgebra × ∪ ran sigAlgebra)
8	7	rexeqi 3320	. . 3 ⊢ (∃𝑎 ∈ dom MblFnM𝐹 ∈ (MblFnM‘𝑎) ↔ ∃𝑎 ∈ (∪ ran sigAlgebra × ∪ ran sigAlgebra)𝐹 ∈ (MblFnM‘𝑎))
9		fveq2 6882	. . . . . 6 ⊢ (𝑎 = ⟨𝑠, 𝑡⟩ → (MblFnM‘𝑎) = (MblFnM‘⟨𝑠, 𝑡⟩))
10		df-ov 7419	. . . . . 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 5826	. . 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 488	. . . 4 ⊢ ((𝑠 ∈ ∪ ran sigAlgebra ∧ 𝑡 ∈ ∪ ran sigAlgebra) → 𝑠 ∈ ∪ ran sigAlgebra)
16		simpr 490	. . . 4 ⊢ ((𝑠 ∈ ∪ ran sigAlgebra ∧ 𝑡 ∈ ∪ ran sigAlgebra) → 𝑡 ∈ ∪ ran sigAlgebra)
17	15, 16	ismbfm 34764	. . 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 401   = wceq 1570   ∈ wcel 2145  ∀wral 3078  ∃wrex 3088  {crab 3414  ⟨cop 4593  ∪ cuni 4870   × cxp 5657  ^◡ccnv 5658  dom cdm 5659  ran crn 5660   “ cima 5662  Fun wfun 6531  ‘cfv 6537  (class class class)co 7416   ↑_m cmap 8829  sigAlgebracsiga 34620  MblFnMcmbfm 34762

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402  ax-un 7739

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  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 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-ov 7419  df-oprab 7420  df-mpo 7421  df-1st 7989  df-2nd 7990  df-mbfm 34763

This theorem is used by:  mbfmfun  34766