Theorem elunirnmbfm 32360

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 32358	. . . . 5 ⊢ MblFnM = (𝑠 ∈ ∪ ran sigAlgebra, 𝑡 ∈ ∪ ran sigAlgebra ↦ {𝑓 ∈ (∪ 𝑡 ↑m ∪ 𝑠) ∣ ∀𝑥 ∈ 𝑡 (◡𝑓 “ 𝑥) ∈ 𝑠})
2	1	mpofun 7440	. . . 4 ⊢ Fun MblFnM
3		elunirn 7164	. . . 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 7350	. . . . . 6 ⊢ (∪ 𝑡 ↑m ∪ 𝑠) ∈ V
6	5	rabex 5271	. . . . 5 ⊢ {𝑓 ∈ (∪ 𝑡 ↑m ∪ 𝑠) ∣ ∀𝑥 ∈ 𝑡 (◡𝑓 “ 𝑥) ∈ 𝑠} ∈ V
7	1, 6	dmmpo 7958	. . . 4 ⊢ dom MblFnM = (∪ ran sigAlgebra × ∪ ran sigAlgebra)
8	7	rexeqi 3309	. . 3 ⊢ (∃𝑎 ∈ dom MblFnM𝐹 ∈ (MblFnM‘𝑎) ↔ ∃𝑎 ∈ (∪ ran sigAlgebra × ∪ ran sigAlgebra)𝐹 ∈ (MblFnM‘𝑎))
9		fveq2 6812	. . . . . 6 ⊢ (𝑎 = ⟨𝑠, 𝑡⟩ → (MblFnM‘𝑎) = (MblFnM‘⟨𝑠, 𝑡⟩))
10		df-ov 7320	. . . . . 6 ⊢ (𝑠MblFnM𝑡) = (MblFnM‘⟨𝑠, 𝑡⟩)
11	9, 10	eqtr4di 2795	. . . . 5 ⊢ (𝑎 = ⟨𝑠, 𝑡⟩ → (MblFnM‘𝑎) = (𝑠MblFnM𝑡))
12	11	eleq2d 2823	. . . 4 ⊢ (𝑎 = ⟨𝑠, 𝑡⟩ → (𝐹 ∈ (MblFnM‘𝑎) ↔ 𝐹 ∈ (𝑠MblFnM𝑡)))
13	12	rexxp 5772	. . 3 ⊢ (∃𝑎 ∈ (∪ ran sigAlgebra × ∪ ran sigAlgebra)𝐹 ∈ (MblFnM‘𝑎) ↔ ∃𝑠 ∈ ∪ ran sigAlgebra∃𝑡 ∈ ∪ ran sigAlgebra𝐹 ∈ (𝑠MblFnM𝑡))
14	4, 8, 13	3bitri 296	. 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 32359	. . 3 ⊢ ((𝑠 ∈ ∪ ran sigAlgebra ∧ 𝑡 ∈ ∪ ran sigAlgebra) → (𝐹 ∈ (𝑠MblFnM𝑡) ↔ (𝐹 ∈ (∪ 𝑡 ↑m ∪ 𝑠) ∧ ∀𝑥 ∈ 𝑡 (◡𝐹 “ 𝑥) ∈ 𝑠)))
18	17	2rexbiia 3206	. 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 1540   ∈ wcel 2105  ∀wral 3062  ∃wrex 3071  {crab 3404  ⟨cop 4577  ∪ cuni 4850   × cxp 5606  ^◡ccnv 5607  dom cdm 5608  ran crn 5609   “ cima 5611  Fun wfun 6460  ‘cfv 6466  (class class class)co 7317   ↑_m cmap 8665  sigAlgebracsiga 32216  MblFnMcmbfm 32357

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2708  ax-sep 5238  ax-nul 5245  ax-pr 5367  ax-un 7630

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3405  df-v 3443  df-sbc 3727  df-csb 3843  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4268  df-if 4472  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4851  df-iun 4939  df-br 5088  df-opab 5150  df-mpt 5171  df-id 5507  df-xp 5614  df-rel 5615  df-cnv 5616  df-co 5617  df-dm 5618  df-rn 5619  df-res 5620  df-ima 5621  df-iota 6418  df-fun 6468  df-fn 6469  df-f 6470  df-fv 6474  df-ov 7320  df-oprab 7321  df-mpo 7322  df-1st 7878  df-2nd 7879  df-mbfm 32358

This theorem is referenced by:  mbfmfun  32361  isanmbfm  32363