Theorem mbfmcnt 34667

Description: All functions are measurable with respect to the counting measure. (Contributed by Thierry Arnoux, 24-Jan-2017.)

Assertion

Ref	Expression
mbfmcnt	⊢ (𝑂 ∈ 𝑉 → (𝒫 𝑂MblFnM𝔅ℝ) = (ℝ ↑m 𝑂))

Proof of Theorem mbfmcnt

Dummy variables 𝑥 𝑓 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pwsiga 34529	. . . . . 6 ⊢ (𝑂 ∈ 𝑉 → 𝒫 𝑂 ∈ (sigAlgebra‘𝑂))
2		elrnsiga 34525	. . . . . 6 ⊢ (𝒫 𝑂 ∈ (sigAlgebra‘𝑂) → 𝒫 𝑂 ∈ ∪ ran sigAlgebra)
3	1, 2	syl 18	. . . . 5 ⊢ (𝑂 ∈ 𝑉 → 𝒫 𝑂 ∈ ∪ ran sigAlgebra)
4		brsigarn 34583	. . . . . 6 ⊢ 𝔅ℝ ∈ (sigAlgebra‘ℝ)
5		elrnsiga 34525	. . . . . 6 ⊢ (𝔅ℝ ∈ (sigAlgebra‘ℝ) → 𝔅ℝ ∈ ∪ ran sigAlgebra)
6	4, 5	mp1i 14	. . . . 5 ⊢ (𝑂 ∈ 𝑉 → 𝔅ℝ ∈ ∪ ran sigAlgebra)
7	3, 6	ismbfm 34650	. . . 4 ⊢ (𝑂 ∈ 𝑉 → (𝑓 ∈ (𝒫 𝑂MblFnM𝔅ℝ) ↔ (𝑓 ∈ (∪ 𝔅ℝ ↑m ∪ 𝒫 𝑂) ∧ ∀𝑥 ∈ 𝔅ℝ (◡𝑓 “ 𝑥) ∈ 𝒫 𝑂)))
8		unibrsiga 34585	. . . . . . . . . 10 ⊢ ∪ 𝔅ℝ = ℝ
9		reex 11197	. . . . . . . . . 10 ⊢ ℝ ∈ V
10	8, 9	eqeltri 2858	. . . . . . . . 9 ⊢ ∪ 𝔅ℝ ∈ V
11		unipw 5430	. . . . . . . . . 10 ⊢ ∪ 𝒫 𝑂 = 𝑂
12		elex 3475	. . . . . . . . . 10 ⊢ (𝑂 ∈ 𝑉 → 𝑂 ∈ V)
13	11, 12	eqeltrid 2866	. . . . . . . . 9 ⊢ (𝑂 ∈ 𝑉 → ∪ 𝒫 𝑂 ∈ V)
14		elmapg 8834	. . . . . . . . 9 ⊢ ((∪ 𝔅ℝ ∈ V ∧ ∪ 𝒫 𝑂 ∈ V) → (𝑓 ∈ (∪ 𝔅ℝ ↑m ∪ 𝒫 𝑂) ↔ 𝑓:∪ 𝒫 𝑂⟶∪ 𝔅ℝ))
15	10, 13, 14	sylancr 598	. . . . . . . 8 ⊢ (𝑂 ∈ 𝑉 → (𝑓 ∈ (∪ 𝔅ℝ ↑m ∪ 𝒫 𝑂) ↔ 𝑓:∪ 𝒫 𝑂⟶∪ 𝔅ℝ))
16	11	feq2i 6697	. . . . . . . 8 ⊢ (𝑓:∪ 𝒫 𝑂⟶∪ 𝔅ℝ ↔ 𝑓:𝑂⟶∪ 𝔅ℝ)
17	15, 16	bitrdi 290	. . . . . . 7 ⊢ (𝑂 ∈ 𝑉 → (𝑓 ∈ (∪ 𝔅ℝ ↑m ∪ 𝒫 𝑂) ↔ 𝑓:𝑂⟶∪ 𝔅ℝ))
18		ffn 6705	. . . . . . 7 ⊢ (𝑓:𝑂⟶∪ 𝔅ℝ → 𝑓 Fn 𝑂)
19	17, 18	biimtrdi 256	. . . . . 6 ⊢ (𝑂 ∈ 𝑉 → (𝑓 ∈ (∪ 𝔅ℝ ↑m ∪ 𝒫 𝑂) → 𝑓 Fn 𝑂))
20		elpreima 7053	. . . . . . . . . 10 ⊢ (𝑓 Fn 𝑂 → (𝑦 ∈ (◡𝑓 “ 𝑥) ↔ (𝑦 ∈ 𝑂 ∧ (𝑓‘𝑦) ∈ 𝑥)))
21		simpl 487	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝑂 ∧ (𝑓‘𝑦) ∈ 𝑥) → 𝑦 ∈ 𝑂)
22	20, 21	biimtrdi 256	. . . . . . . . 9 ⊢ (𝑓 Fn 𝑂 → (𝑦 ∈ (◡𝑓 “ 𝑥) → 𝑦 ∈ 𝑂))
23	22	ssrdv 3942	. . . . . . . 8 ⊢ (𝑓 Fn 𝑂 → (◡𝑓 “ 𝑥) ⊆ 𝑂)
24		vex 3458	. . . . . . . . . . 11 ⊢ 𝑓 ∈ V
25	24	cnvex 7920	. . . . . . . . . 10 ⊢ ◡𝑓 ∈ V
26		imaexg 7908	. . . . . . . . . 10 ⊢ (◡𝑓 ∈ V → (◡𝑓 “ 𝑥) ∈ V)
27	25, 26	ax-mp 5	. . . . . . . . 9 ⊢ (◡𝑓 “ 𝑥) ∈ V
28	27	elpw 4565	. . . . . . . 8 ⊢ ((◡𝑓 “ 𝑥) ∈ 𝒫 𝑂 ↔ (◡𝑓 “ 𝑥) ⊆ 𝑂)
29	23, 28	sylibr 237	. . . . . . 7 ⊢ (𝑓 Fn 𝑂 → (◡𝑓 “ 𝑥) ∈ 𝒫 𝑂)
30	29	ralrimivw 3160	. . . . . 6 ⊢ (𝑓 Fn 𝑂 → ∀𝑥 ∈ 𝔅ℝ (◡𝑓 “ 𝑥) ∈ 𝒫 𝑂)
31	19, 30	syl6 36	. . . . 5 ⊢ (𝑂 ∈ 𝑉 → (𝑓 ∈ (∪ 𝔅ℝ ↑m ∪ 𝒫 𝑂) → ∀𝑥 ∈ 𝔅ℝ (◡𝑓 “ 𝑥) ∈ 𝒫 𝑂))
32	31	pm4.71d 570	. . . 4 ⊢ (𝑂 ∈ 𝑉 → (𝑓 ∈ (∪ 𝔅ℝ ↑m ∪ 𝒫 𝑂) ↔ (𝑓 ∈ (∪ 𝔅ℝ ↑m ∪ 𝒫 𝑂) ∧ ∀𝑥 ∈ 𝔅ℝ (◡𝑓 “ 𝑥) ∈ 𝒫 𝑂)))
33	7, 32	bitr4d 285	. . 3 ⊢ (𝑂 ∈ 𝑉 → (𝑓 ∈ (𝒫 𝑂MblFnM𝔅ℝ) ↔ 𝑓 ∈ (∪ 𝔅ℝ ↑m ∪ 𝒫 𝑂)))
34	33	eqrdv 2760	. 2 ⊢ (𝑂 ∈ 𝑉 → (𝒫 𝑂MblFnM𝔅ℝ) = (∪ 𝔅ℝ ↑m ∪ 𝒫 𝑂))
35	8, 11	oveq12i 7424	. 2 ⊢ (∪ 𝔅ℝ ↑m ∪ 𝒫 𝑂) = (ℝ ↑m 𝑂)
36	34, 35	eqtrdi 2813	1 ⊢ (𝑂 ∈ 𝑉 → (𝒫 𝑂MblFnM𝔅ℝ) = (ℝ ↑m 𝑂))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1569   ∈ wcel 2142  ∀wral 3078  Vcvv 3454   ⊆ wss 3904  𝒫 cpw 4561  ∪ cuni 4871  ^◡ccnv 5659  ran crn 5661   “ cima 5663   Fn wfn 6531  ⟶wf 6532  ‘cfv 6536  (class class class)co 7412   ↑_m cmap 8822  ℝcr 11105  sigAlgebracsiga 34507  𝔅_ℝcbrsiga 34580  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-pow 5335  ax-pr 5403  ax-un 7734  ax-cnex 11162  ax-resscn 11163  ax-pre-lttri 11180  ax-pre-lttrn 11181

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1103  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-nel 3064  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-int 4912  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5555  df-po 5568  df-so 5569  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-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7984  df-2nd 7985  df-er 8692  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-pnf 11251  df-mnf 11252  df-xr 11253  df-ltxr 11254  df-le 11255  df-ioo 13382  df-topgen 17502  df-top 23062  df-bases 23114  df-siga 34508  df-sigagen 34538  df-brsiga 34581  df-mbfm 34649

This theorem is used by: (None)