Theorem mbfmcnt 34820

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 34681	. . . . . 6 ⊢ (𝑂 ∈ 𝑉 → 𝒫 𝑂 ∈ (sigAlgebra‘𝑂))
2		elrnsiga 34677	. . . . . 6 ⊢ (𝒫 𝑂 ∈ (sigAlgebra‘𝑂) → 𝒫 𝑂 ∈ ∪ ran sigAlgebra)
3	1, 2	syl 18	. . . . 5 ⊢ (𝑂 ∈ 𝑉 → 𝒫 𝑂 ∈ ∪ ran sigAlgebra)
4		brsigarn 34736	. . . . . 6 ⊢ 𝔅ℝ ∈ (sigAlgebra‘ℝ)
5		elrnsiga 34677	. . . . . 6 ⊢ (𝔅ℝ ∈ (sigAlgebra‘ℝ) → 𝔅ℝ ∈ ∪ ran sigAlgebra)
6	4, 5	mp1i 14	. . . . 5 ⊢ (𝑂 ∈ 𝑉 → 𝔅ℝ ∈ ∪ ran sigAlgebra)
7	3, 6	ismbfm 34803	. . . 4 ⊢ (𝑂 ∈ 𝑉 → (𝑓 ∈ (𝒫 𝑂MblFnM𝔅ℝ) ↔ (𝑓 ∈ (∪ 𝔅ℝ ↑m ∪ 𝒫 𝑂) ∧ ∀𝑥 ∈ 𝔅ℝ (◡𝑓 “ 𝑥) ∈ 𝒫 𝑂)))
8		unibrsiga 34738	. . . . . . . . . 10 ⊢ ∪ 𝔅ℝ = ℝ
9		reex 11248	. . . . . . . . . 10 ⊢ ℝ ∈ V
10	8, 9	eqeltri 2856	. . . . . . . . 9 ⊢ ∪ 𝔅ℝ ∈ V
11		unipw 5418	. . . . . . . . . 10 ⊢ ∪ 𝒫 𝑂 = 𝑂
12		elex 3471	. . . . . . . . . 10 ⊢ (𝑂 ∈ 𝑉 → 𝑂 ∈ V)
13	11, 12	eqeltrid 2864	. . . . . . . . 9 ⊢ (𝑂 ∈ 𝑉 → ∪ 𝒫 𝑂 ∈ V)
14		elmapg 8838	. . . . . . . . 9 ⊢ ((∪ 𝔅ℝ ∈ V ∧ ∪ 𝒫 𝑂 ∈ V) → (𝑓 ∈ (∪ 𝔅ℝ ↑m ∪ 𝒫 𝑂) ↔ 𝑓:∪ 𝒫 𝑂⟶∪ 𝔅ℝ))
15	10, 13, 14	sylancr 599	. . . . . . . 8 ⊢ (𝑂 ∈ 𝑉 → (𝑓 ∈ (∪ 𝔅ℝ ↑m ∪ 𝒫 𝑂) ↔ 𝑓:∪ 𝒫 𝑂⟶∪ 𝔅ℝ))
16	11	feq2i 6690	. . . . . . . 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 6698	. . . . . . 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 7046	. . . . . . . . . 10 ⊢ (𝑓 Fn 𝑂 → (𝑦 ∈ (◡𝑓 “ 𝑥) ↔ (𝑦 ∈ 𝑂 ∧ (𝑓‘𝑦) ∈ 𝑥)))
21		simpl 488	. . . . . . . . . 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 3937	. . . . . . . 8 ⊢ (𝑓 Fn 𝑂 → (◡𝑓 “ 𝑥) ⊆ 𝑂)
24		vex 3454	. . . . . . . . . . 11 ⊢ 𝑓 ∈ V
25	24	cnvex 7921	. . . . . . . . . 10 ⊢ ◡𝑓 ∈ V
26		imaexg 7909	. . . . . . . . . 10 ⊢ (◡𝑓 ∈ V → (◡𝑓 “ 𝑥) ∈ V)
27	25, 26	ax-mp 5	. . . . . . . . 9 ⊢ (◡𝑓 “ 𝑥) ∈ V
28	27	elpw 4561	. . . . . . . 8 ⊢ ((◡𝑓 “ 𝑥) ∈ 𝒫 𝑂 ↔ (◡𝑓 “ 𝑥) ⊆ 𝑂)
29	23, 28	sylibr 237	. . . . . . 7 ⊢ (𝑓 Fn 𝑂 → (◡𝑓 “ 𝑥) ∈ 𝒫 𝑂)
30	29	ralrimivw 3158	. . . . . 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 571	. . . 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 2758	. 2 ⊢ (𝑂 ∈ 𝑉 → (𝒫 𝑂MblFnM𝔅ℝ) = (∪ 𝔅ℝ ↑m ∪ 𝒫 𝑂))
35	8, 11	oveq12i 7421	. 2 ⊢ (∪ 𝔅ℝ ↑m ∪ 𝒫 𝑂) = (ℝ ↑m 𝑂)
36	34, 35	eqtrdi 2811	1 ⊢ (𝑂 ∈ 𝑉 → (𝒫 𝑂MblFnM𝔅ℝ) = (ℝ ↑m 𝑂))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3076  Vcvv 3450   ⊆ wss 3899  𝒫 cpw 4557  ∪ cuni 4867  ^◡ccnv 5647  ran crn 5649   “ cima 5651   Fn wfn 6523  ⟶wf 6524  ‘cfv 6528  (class class class)co 7409   ↑_m cmap 8826  ℝcr 11156  sigAlgebracsiga 34659  𝔅_ℝcbrsiga 34733  MblFnMcmbfm 34801

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 2213  ax-ext 2732  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7735  ax-cnex 11213  ax-resscn 11214  ax-pre-lttri 11231  ax-pre-lttrn 11232

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5543  df-po 5556  df-so 5557  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-iota 6484  df-fun 6530  df-fn 6531  df-f 6532  df-f1 6533  df-fo 6534  df-f1o 6535  df-fv 6536  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7985  df-2nd 7986  df-er 8696  df-map 8828  df-en 8953  df-dom 8954  df-sdom 8955  df-pnf 11302  df-mnf 11303  df-xr 11304  df-ltxr 11305  df-le 11306  df-ioo 13435  df-topgen 17561  df-top 23159  df-bases 23211  df-siga 34660  df-sigagen 34691  df-brsiga 34734  df-mbfm 34802

This theorem is used by: (None)