Theorem mbfmcnt 32135

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 31998	. . . . . 6 ⊢ (𝑂 ∈ 𝑉 → 𝒫 𝑂 ∈ (sigAlgebra‘𝑂))
2		elrnsiga 31994	. . . . . 6 ⊢ (𝒫 𝑂 ∈ (sigAlgebra‘𝑂) → 𝒫 𝑂 ∈ ∪ ran sigAlgebra)
3	1, 2	syl 17	. . . . 5 ⊢ (𝑂 ∈ 𝑉 → 𝒫 𝑂 ∈ ∪ ran sigAlgebra)
4		brsigarn 32052	. . . . . 6 ⊢ 𝔅ℝ ∈ (sigAlgebra‘ℝ)
5		elrnsiga 31994	. . . . . 6 ⊢ (𝔅ℝ ∈ (sigAlgebra‘ℝ) → 𝔅ℝ ∈ ∪ ran sigAlgebra)
6	4, 5	mp1i 13	. . . . 5 ⊢ (𝑂 ∈ 𝑉 → 𝔅ℝ ∈ ∪ ran sigAlgebra)
7	3, 6	ismbfm 32119	. . . 4 ⊢ (𝑂 ∈ 𝑉 → (𝑓 ∈ (𝒫 𝑂MblFnM𝔅ℝ) ↔ (𝑓 ∈ (∪ 𝔅ℝ ↑m ∪ 𝒫 𝑂) ∧ ∀𝑥 ∈ 𝔅ℝ (◡𝑓 “ 𝑥) ∈ 𝒫 𝑂)))
8		unibrsiga 32054	. . . . . . . . . 10 ⊢ ∪ 𝔅ℝ = ℝ
9		reex 10893	. . . . . . . . . 10 ⊢ ℝ ∈ V
10	8, 9	eqeltri 2835	. . . . . . . . 9 ⊢ ∪ 𝔅ℝ ∈ V
11		unipw 5360	. . . . . . . . . 10 ⊢ ∪ 𝒫 𝑂 = 𝑂
12		elex 3440	. . . . . . . . . 10 ⊢ (𝑂 ∈ 𝑉 → 𝑂 ∈ V)
13	11, 12	eqeltrid 2843	. . . . . . . . 9 ⊢ (𝑂 ∈ 𝑉 → ∪ 𝒫 𝑂 ∈ V)
14		elmapg 8586	. . . . . . . . 9 ⊢ ((∪ 𝔅ℝ ∈ V ∧ ∪ 𝒫 𝑂 ∈ V) → (𝑓 ∈ (∪ 𝔅ℝ ↑m ∪ 𝒫 𝑂) ↔ 𝑓:∪ 𝒫 𝑂⟶∪ 𝔅ℝ))
15	10, 13, 14	sylancr 586	. . . . . . . 8 ⊢ (𝑂 ∈ 𝑉 → (𝑓 ∈ (∪ 𝔅ℝ ↑m ∪ 𝒫 𝑂) ↔ 𝑓:∪ 𝒫 𝑂⟶∪ 𝔅ℝ))
16	11	feq2i 6576	. . . . . . . 8 ⊢ (𝑓:∪ 𝒫 𝑂⟶∪ 𝔅ℝ ↔ 𝑓:𝑂⟶∪ 𝔅ℝ)
17	15, 16	bitrdi 286	. . . . . . 7 ⊢ (𝑂 ∈ 𝑉 → (𝑓 ∈ (∪ 𝔅ℝ ↑m ∪ 𝒫 𝑂) ↔ 𝑓:𝑂⟶∪ 𝔅ℝ))
18		ffn 6584	. . . . . . 7 ⊢ (𝑓:𝑂⟶∪ 𝔅ℝ → 𝑓 Fn 𝑂)
19	17, 18	syl6bi 252	. . . . . 6 ⊢ (𝑂 ∈ 𝑉 → (𝑓 ∈ (∪ 𝔅ℝ ↑m ∪ 𝒫 𝑂) → 𝑓 Fn 𝑂))
20		elpreima 6917	. . . . . . . . . 10 ⊢ (𝑓 Fn 𝑂 → (𝑦 ∈ (◡𝑓 “ 𝑥) ↔ (𝑦 ∈ 𝑂 ∧ (𝑓‘𝑦) ∈ 𝑥)))
21		simpl 482	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝑂 ∧ (𝑓‘𝑦) ∈ 𝑥) → 𝑦 ∈ 𝑂)
22	20, 21	syl6bi 252	. . . . . . . . 9 ⊢ (𝑓 Fn 𝑂 → (𝑦 ∈ (◡𝑓 “ 𝑥) → 𝑦 ∈ 𝑂))
23	22	ssrdv 3923	. . . . . . . 8 ⊢ (𝑓 Fn 𝑂 → (◡𝑓 “ 𝑥) ⊆ 𝑂)
24		vex 3426	. . . . . . . . . . 11 ⊢ 𝑓 ∈ V
25	24	cnvex 7746	. . . . . . . . . 10 ⊢ ◡𝑓 ∈ V
26		imaexg 7736	. . . . . . . . . 10 ⊢ (◡𝑓 ∈ V → (◡𝑓 “ 𝑥) ∈ V)
27	25, 26	ax-mp 5	. . . . . . . . 9 ⊢ (◡𝑓 “ 𝑥) ∈ V
28	27	elpw 4534	. . . . . . . 8 ⊢ ((◡𝑓 “ 𝑥) ∈ 𝒫 𝑂 ↔ (◡𝑓 “ 𝑥) ⊆ 𝑂)
29	23, 28	sylibr 233	. . . . . . 7 ⊢ (𝑓 Fn 𝑂 → (◡𝑓 “ 𝑥) ∈ 𝒫 𝑂)
30	29	ralrimivw 3108	. . . . . 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 561	. . . 4 ⊢ (𝑂 ∈ 𝑉 → (𝑓 ∈ (∪ 𝔅ℝ ↑m ∪ 𝒫 𝑂) ↔ (𝑓 ∈ (∪ 𝔅ℝ ↑m ∪ 𝒫 𝑂) ∧ ∀𝑥 ∈ 𝔅ℝ (◡𝑓 “ 𝑥) ∈ 𝒫 𝑂)))
33	7, 32	bitr4d 281	. . 3 ⊢ (𝑂 ∈ 𝑉 → (𝑓 ∈ (𝒫 𝑂MblFnM𝔅ℝ) ↔ 𝑓 ∈ (∪ 𝔅ℝ ↑m ∪ 𝒫 𝑂)))
34	33	eqrdv 2736	. 2 ⊢ (𝑂 ∈ 𝑉 → (𝒫 𝑂MblFnM𝔅ℝ) = (∪ 𝔅ℝ ↑m ∪ 𝒫 𝑂))
35	8, 11	oveq12i 7267	. 2 ⊢ (∪ 𝔅ℝ ↑m ∪ 𝒫 𝑂) = (ℝ ↑m 𝑂)
36	34, 35	eqtrdi 2795	1 ⊢ (𝑂 ∈ 𝑉 → (𝒫 𝑂MblFnM𝔅ℝ) = (ℝ ↑m 𝑂))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∀wral 3063  Vcvv 3422   ⊆ wss 3883  𝒫 cpw 4530  ∪ cuni 4836  ^◡ccnv 5579  ran crn 5581   “ cima 5583   Fn wfn 6413  ⟶wf 6414  ‘cfv 6418  (class class class)co 7255   ↑_m cmap 8573  ℝcr 10801  sigAlgebracsiga 31976  𝔅_ℝcbrsiga 32049  MblFnMcmbfm 32117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-pre-lttri 10876  ax-pre-lttrn 10877

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-po 5494  df-so 5495  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-1st 7804  df-2nd 7805  df-er 8456  df-map 8575  df-en 8692  df-dom 8693  df-sdom 8694  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-ioo 13012  df-topgen 17071  df-top 21951  df-bases 22004  df-siga 31977  df-sigagen 32007  df-brsiga 32050  df-mbfm 32118

This theorem is referenced by: (None)