Theorem mbfmcnt 31636

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 31499	. . . . . 6 ⊢ (𝑂 ∈ 𝑉 → 𝒫 𝑂 ∈ (sigAlgebra‘𝑂))
2		elrnsiga 31495	. . . . . 6 ⊢ (𝒫 𝑂 ∈ (sigAlgebra‘𝑂) → 𝒫 𝑂 ∈ ∪ ran sigAlgebra)
3	1, 2	syl 17	. . . . 5 ⊢ (𝑂 ∈ 𝑉 → 𝒫 𝑂 ∈ ∪ ran sigAlgebra)
4		brsigarn 31553	. . . . . 6 ⊢ 𝔅ℝ ∈ (sigAlgebra‘ℝ)
5		elrnsiga 31495	. . . . . 6 ⊢ (𝔅ℝ ∈ (sigAlgebra‘ℝ) → 𝔅ℝ ∈ ∪ ran sigAlgebra)
6	4, 5	mp1i 13	. . . . 5 ⊢ (𝑂 ∈ 𝑉 → 𝔅ℝ ∈ ∪ ran sigAlgebra)
7	3, 6	ismbfm 31620	. . . 4 ⊢ (𝑂 ∈ 𝑉 → (𝑓 ∈ (𝒫 𝑂MblFnM𝔅ℝ) ↔ (𝑓 ∈ (∪ 𝔅ℝ ↑m ∪ 𝒫 𝑂) ∧ ∀𝑥 ∈ 𝔅ℝ (◡𝑓 “ 𝑥) ∈ 𝒫 𝑂)))
8		unibrsiga 31555	. . . . . . . . . 10 ⊢ ∪ 𝔅ℝ = ℝ
9		reex 10617	. . . . . . . . . 10 ⊢ ℝ ∈ V
10	8, 9	eqeltri 2886	. . . . . . . . 9 ⊢ ∪ 𝔅ℝ ∈ V
11		unipw 5308	. . . . . . . . . 10 ⊢ ∪ 𝒫 𝑂 = 𝑂
12		elex 3459	. . . . . . . . . 10 ⊢ (𝑂 ∈ 𝑉 → 𝑂 ∈ V)
13	11, 12	eqeltrid 2894	. . . . . . . . 9 ⊢ (𝑂 ∈ 𝑉 → ∪ 𝒫 𝑂 ∈ V)
14		elmapg 8402	. . . . . . . . 9 ⊢ ((∪ 𝔅ℝ ∈ V ∧ ∪ 𝒫 𝑂 ∈ V) → (𝑓 ∈ (∪ 𝔅ℝ ↑m ∪ 𝒫 𝑂) ↔ 𝑓:∪ 𝒫 𝑂⟶∪ 𝔅ℝ))
15	10, 13, 14	sylancr 590	. . . . . . . 8 ⊢ (𝑂 ∈ 𝑉 → (𝑓 ∈ (∪ 𝔅ℝ ↑m ∪ 𝒫 𝑂) ↔ 𝑓:∪ 𝒫 𝑂⟶∪ 𝔅ℝ))
16	11	feq2i 6479	. . . . . . . 8 ⊢ (𝑓:∪ 𝒫 𝑂⟶∪ 𝔅ℝ ↔ 𝑓:𝑂⟶∪ 𝔅ℝ)
17	15, 16	syl6bb 290	. . . . . . 7 ⊢ (𝑂 ∈ 𝑉 → (𝑓 ∈ (∪ 𝔅ℝ ↑m ∪ 𝒫 𝑂) ↔ 𝑓:𝑂⟶∪ 𝔅ℝ))
18		ffn 6487	. . . . . . 7 ⊢ (𝑓:𝑂⟶∪ 𝔅ℝ → 𝑓 Fn 𝑂)
19	17, 18	syl6bi 256	. . . . . 6 ⊢ (𝑂 ∈ 𝑉 → (𝑓 ∈ (∪ 𝔅ℝ ↑m ∪ 𝒫 𝑂) → 𝑓 Fn 𝑂))
20		elpreima 6805	. . . . . . . . . 10 ⊢ (𝑓 Fn 𝑂 → (𝑦 ∈ (◡𝑓 “ 𝑥) ↔ (𝑦 ∈ 𝑂 ∧ (𝑓‘𝑦) ∈ 𝑥)))
21		simpl 486	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝑂 ∧ (𝑓‘𝑦) ∈ 𝑥) → 𝑦 ∈ 𝑂)
22	20, 21	syl6bi 256	. . . . . . . . 9 ⊢ (𝑓 Fn 𝑂 → (𝑦 ∈ (◡𝑓 “ 𝑥) → 𝑦 ∈ 𝑂))
23	22	ssrdv 3921	. . . . . . . 8 ⊢ (𝑓 Fn 𝑂 → (◡𝑓 “ 𝑥) ⊆ 𝑂)
24		vex 3444	. . . . . . . . . . 11 ⊢ 𝑓 ∈ V
25	24	cnvex 7612	. . . . . . . . . 10 ⊢ ◡𝑓 ∈ V
26		imaexg 7602	. . . . . . . . . 10 ⊢ (◡𝑓 ∈ V → (◡𝑓 “ 𝑥) ∈ V)
27	25, 26	ax-mp 5	. . . . . . . . 9 ⊢ (◡𝑓 “ 𝑥) ∈ V
28	27	elpw 4501	. . . . . . . 8 ⊢ ((◡𝑓 “ 𝑥) ∈ 𝒫 𝑂 ↔ (◡𝑓 “ 𝑥) ⊆ 𝑂)
29	23, 28	sylibr 237	. . . . . . 7 ⊢ (𝑓 Fn 𝑂 → (◡𝑓 “ 𝑥) ∈ 𝒫 𝑂)
30	29	ralrimivw 3150	. . . . . 6 ⊢ (𝑓 Fn 𝑂 → ∀𝑥 ∈ 𝔅ℝ (◡𝑓 “ 𝑥) ∈ 𝒫 𝑂)
31	19, 30	syl6 35	. . . . 5 ⊢ (𝑂 ∈ 𝑉 → (𝑓 ∈ (∪ 𝔅ℝ ↑m ∪ 𝒫 𝑂) → ∀𝑥 ∈ 𝔅ℝ (◡𝑓 “ 𝑥) ∈ 𝒫 𝑂))
32	31	pm4.71d 565	. . . 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 2796	. 2 ⊢ (𝑂 ∈ 𝑉 → (𝒫 𝑂MblFnM𝔅ℝ) = (∪ 𝔅ℝ ↑m ∪ 𝒫 𝑂))
35	8, 11	oveq12i 7147	. 2 ⊢ (∪ 𝔅ℝ ↑m ∪ 𝒫 𝑂) = (ℝ ↑m 𝑂)
36	34, 35	eqtrdi 2849	1 ⊢ (𝑂 ∈ 𝑉 → (𝒫 𝑂MblFnM𝔅ℝ) = (ℝ ↑m 𝑂))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106  Vcvv 3441   ⊆ wss 3881  𝒫 cpw 4497  ∪ cuni 4800  ^◡ccnv 5518  ran crn 5520   “ cima 5522   Fn wfn 6319  ⟶wf 6320  ‘cfv 6324  (class class class)co 7135   ↑_m cmap 8389  ℝcr 10525  sigAlgebracsiga 31477  𝔅_ℝcbrsiga 31550  MblFnMcmbfm 31618

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-cnex 10582  ax-resscn 10583  ax-pre-lttri 10600  ax-pre-lttrn 10601

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-po 5438  df-so 5439  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-1st 7671  df-2nd 7672  df-er 8272  df-map 8391  df-en 8493  df-dom 8494  df-sdom 8495  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-ioo 12730  df-topgen 16709  df-top 21499  df-bases 21551  df-siga 31478  df-sigagen 31508  df-brsiga 31551  df-mbfm 31619

This theorem is referenced by: (None)