Theorem mbfmcnt 30330

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 30193	. . . . . 6 ⊢ (𝑂 ∈ 𝑉 → 𝒫 𝑂 ∈ (sigAlgebra‘𝑂))
2		elrnsiga 30189	. . . . . 6 ⊢ (𝒫 𝑂 ∈ (sigAlgebra‘𝑂) → 𝒫 𝑂 ∈ ∪ ran sigAlgebra)
3	1, 2	syl 17	. . . . 5 ⊢ (𝑂 ∈ 𝑉 → 𝒫 𝑂 ∈ ∪ ran sigAlgebra)
4		brsigarn 30247	. . . . . 6 ⊢ 𝔅ℝ ∈ (sigAlgebra‘ℝ)
5		elrnsiga 30189	. . . . . 6 ⊢ (𝔅ℝ ∈ (sigAlgebra‘ℝ) → 𝔅ℝ ∈ ∪ ran sigAlgebra)
6	4, 5	mp1i 13	. . . . 5 ⊢ (𝑂 ∈ 𝑉 → 𝔅ℝ ∈ ∪ ran sigAlgebra)
7	3, 6	ismbfm 30314	. . . 4 ⊢ (𝑂 ∈ 𝑉 → (𝑓 ∈ (𝒫 𝑂MblFnM𝔅ℝ) ↔ (𝑓 ∈ (∪ 𝔅ℝ ↑𝑚 ∪ 𝒫 𝑂) ∧ ∀𝑥 ∈ 𝔅ℝ (◡𝑓 “ 𝑥) ∈ 𝒫 𝑂)))
8		unibrsiga 30249	. . . . . . . . . 10 ⊢ ∪ 𝔅ℝ = ℝ
9		reex 10027	. . . . . . . . . 10 ⊢ ℝ ∈ V
10	8, 9	eqeltri 2697	. . . . . . . . 9 ⊢ ∪ 𝔅ℝ ∈ V
11		unipw 4918	. . . . . . . . . 10 ⊢ ∪ 𝒫 𝑂 = 𝑂
12		elex 3212	. . . . . . . . . 10 ⊢ (𝑂 ∈ 𝑉 → 𝑂 ∈ V)
13	11, 12	syl5eqel 2705	. . . . . . . . 9 ⊢ (𝑂 ∈ 𝑉 → ∪ 𝒫 𝑂 ∈ V)
14		elmapg 7870	. . . . . . . . 9 ⊢ ((∪ 𝔅ℝ ∈ V ∧ ∪ 𝒫 𝑂 ∈ V) → (𝑓 ∈ (∪ 𝔅ℝ ↑𝑚 ∪ 𝒫 𝑂) ↔ 𝑓:∪ 𝒫 𝑂⟶∪ 𝔅ℝ))
15	10, 13, 14	sylancr 695	. . . . . . . 8 ⊢ (𝑂 ∈ 𝑉 → (𝑓 ∈ (∪ 𝔅ℝ ↑𝑚 ∪ 𝒫 𝑂) ↔ 𝑓:∪ 𝒫 𝑂⟶∪ 𝔅ℝ))
16	11	feq2i 6037	. . . . . . . 8 ⊢ (𝑓:∪ 𝒫 𝑂⟶∪ 𝔅ℝ ↔ 𝑓:𝑂⟶∪ 𝔅ℝ)
17	15, 16	syl6bb 276	. . . . . . 7 ⊢ (𝑂 ∈ 𝑉 → (𝑓 ∈ (∪ 𝔅ℝ ↑𝑚 ∪ 𝒫 𝑂) ↔ 𝑓:𝑂⟶∪ 𝔅ℝ))
18		ffn 6045	. . . . . . 7 ⊢ (𝑓:𝑂⟶∪ 𝔅ℝ → 𝑓 Fn 𝑂)
19	17, 18	syl6bi 243	. . . . . 6 ⊢ (𝑂 ∈ 𝑉 → (𝑓 ∈ (∪ 𝔅ℝ ↑𝑚 ∪ 𝒫 𝑂) → 𝑓 Fn 𝑂))
20		elpreima 6337	. . . . . . . . . 10 ⊢ (𝑓 Fn 𝑂 → (𝑦 ∈ (◡𝑓 “ 𝑥) ↔ (𝑦 ∈ 𝑂 ∧ (𝑓‘𝑦) ∈ 𝑥)))
21		simpl 473	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝑂 ∧ (𝑓‘𝑦) ∈ 𝑥) → 𝑦 ∈ 𝑂)
22	20, 21	syl6bi 243	. . . . . . . . 9 ⊢ (𝑓 Fn 𝑂 → (𝑦 ∈ (◡𝑓 “ 𝑥) → 𝑦 ∈ 𝑂))
23	22	ssrdv 3609	. . . . . . . 8 ⊢ (𝑓 Fn 𝑂 → (◡𝑓 “ 𝑥) ⊆ 𝑂)
24		vex 3203	. . . . . . . . . . 11 ⊢ 𝑓 ∈ V
25	24	cnvex 7113	. . . . . . . . . 10 ⊢ ◡𝑓 ∈ V
26		imaexg 7103	. . . . . . . . . 10 ⊢ (◡𝑓 ∈ V → (◡𝑓 “ 𝑥) ∈ V)
27	25, 26	ax-mp 5	. . . . . . . . 9 ⊢ (◡𝑓 “ 𝑥) ∈ V
28	27	elpw 4164	. . . . . . . 8 ⊢ ((◡𝑓 “ 𝑥) ∈ 𝒫 𝑂 ↔ (◡𝑓 “ 𝑥) ⊆ 𝑂)
29	23, 28	sylibr 224	. . . . . . 7 ⊢ (𝑓 Fn 𝑂 → (◡𝑓 “ 𝑥) ∈ 𝒫 𝑂)
30	29	ralrimivw 2967	. . . . . 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 666	. . . 4 ⊢ (𝑂 ∈ 𝑉 → (𝑓 ∈ (∪ 𝔅ℝ ↑𝑚 ∪ 𝒫 𝑂) ↔ (𝑓 ∈ (∪ 𝔅ℝ ↑𝑚 ∪ 𝒫 𝑂) ∧ ∀𝑥 ∈ 𝔅ℝ (◡𝑓 “ 𝑥) ∈ 𝒫 𝑂)))
33	7, 32	bitr4d 271	. . 3 ⊢ (𝑂 ∈ 𝑉 → (𝑓 ∈ (𝒫 𝑂MblFnM𝔅ℝ) ↔ 𝑓 ∈ (∪ 𝔅ℝ ↑𝑚 ∪ 𝒫 𝑂)))
34	33	eqrdv 2620	. 2 ⊢ (𝑂 ∈ 𝑉 → (𝒫 𝑂MblFnM𝔅ℝ) = (∪ 𝔅ℝ ↑𝑚 ∪ 𝒫 𝑂))
35	8, 11	oveq12i 6662	. 2 ⊢ (∪ 𝔅ℝ ↑𝑚 ∪ 𝒫 𝑂) = (ℝ ↑𝑚 𝑂)
36	34, 35	syl6eq 2672	1 ⊢ (𝑂 ∈ 𝑉 → (𝒫 𝑂MblFnM𝔅ℝ) = (ℝ ↑𝑚 𝑂))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  Vcvv 3200   ⊆ wss 3574  𝒫 cpw 4158  ∪ cuni 4436  ^◡ccnv 5113  ran crn 5115   “ cima 5117   Fn wfn 5883  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650   ↑_𝑚 cmap 7857  ℝcr 9935  sigAlgebracsiga 30170  𝔅_ℝcbrsiga 30244  MblFnMcmbfm 30312

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-pre-lttri 10010  ax-pre-lttrn 10011

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-ioo 12179  df-topgen 16104  df-top 20699  df-bases 20750  df-siga 30171  df-sigagen 30202  df-brsiga 30245  df-mbfm 30313

This theorem is referenced by: (None)