Mathbox for Glauco Siliprandi < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  df-smblfn Structured version   Visualization version   GIF version

Definition df-smblfn 42972
 Description: Define a real-valued measurable function w.r.t. a given sigma-algebra. See Definition 121C of [Fremlin1] p. 36 and Definition 135E (b) of [Fremlin1] p. 80 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Assertion
Ref Expression
df-smblfn SMblFn = (𝑠 ∈ SAlg ↦ {𝑓 ∈ (ℝ ↑pm 𝑠) ∣ ∀𝑎 ∈ ℝ (𝑓 “ (-∞(,)𝑎)) ∈ (𝑠t dom 𝑓)})
Distinct variable group:   𝑓,𝑠,𝑎

Detailed syntax breakdown of Definition df-smblfn
StepHypRef Expression
1 csmblfn 42971 . 2 class SMblFn
2 vs . . 3 setvar 𝑠
3 csalg 42587 . . 3 class SAlg
4 vf . . . . . . . . 9 setvar 𝑓
54cv 1532 . . . . . . . 8 class 𝑓
65ccnv 5548 . . . . . . 7 class 𝑓
7 cmnf 10667 . . . . . . . 8 class -∞
8 va . . . . . . . . 9 setvar 𝑎
98cv 1532 . . . . . . . 8 class 𝑎
10 cioo 12732 . . . . . . . 8 class (,)
117, 9, 10co 7150 . . . . . . 7 class (-∞(,)𝑎)
126, 11cima 5552 . . . . . 6 class (𝑓 “ (-∞(,)𝑎))
132cv 1532 . . . . . . 7 class 𝑠
145cdm 5549 . . . . . . 7 class dom 𝑓
15 crest 16688 . . . . . . 7 class t
1613, 14, 15co 7150 . . . . . 6 class (𝑠t dom 𝑓)
1712, 16wcel 2110 . . . . 5 wff (𝑓 “ (-∞(,)𝑎)) ∈ (𝑠t dom 𝑓)
18 cr 10530 . . . . 5 class
1917, 8, 18wral 3138 . . . 4 wff 𝑎 ∈ ℝ (𝑓 “ (-∞(,)𝑎)) ∈ (𝑠t dom 𝑓)
2013cuni 4831 . . . . 5 class 𝑠
21 cpm 8401 . . . . 5 class pm
2218, 20, 21co 7150 . . . 4 class (ℝ ↑pm 𝑠)
2319, 4, 22crab 3142 . . 3 class {𝑓 ∈ (ℝ ↑pm 𝑠) ∣ ∀𝑎 ∈ ℝ (𝑓 “ (-∞(,)𝑎)) ∈ (𝑠t dom 𝑓)}
242, 3, 23cmpt 5138 . 2 class (𝑠 ∈ SAlg ↦ {𝑓 ∈ (ℝ ↑pm 𝑠) ∣ ∀𝑎 ∈ ℝ (𝑓 “ (-∞(,)𝑎)) ∈ (𝑠t dom 𝑓)})
251, 24wceq 1533 1 wff SMblFn = (𝑠 ∈ SAlg ↦ {𝑓 ∈ (ℝ ↑pm 𝑠) ∣ ∀𝑎 ∈ ℝ (𝑓 “ (-∞(,)𝑎)) ∈ (𝑠t dom 𝑓)})
 Colors of variables: wff setvar class This definition is referenced by:  issmflem  42998
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