Definition df-smblfn 45412

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)) ∈ (𝑠 ↾t dom 𝑓)})

Distinct variable group:   𝑓,𝑠,𝑎

Detailed syntax breakdown of Definition df-smblfn

Step	Hyp	Ref	Expression
1		csmblfn 45411	. 2 class SMblFn
2		vs	. . 3 setvar 𝑠
3		csalg 45024	. . 3 class SAlg
4		vf	. . . . . . . . 9 setvar 𝑓
5	4	cv 1541	. . . . . . . 8 class 𝑓
6	5	ccnv 5676	. . . . . . 7 class ◡𝑓
7		cmnf 11246	. . . . . . . 8 class -∞
8		va	. . . . . . . . 9 setvar 𝑎
9	8	cv 1541	. . . . . . . 8 class 𝑎
10		cioo 13324	. . . . . . . 8 class (,)
11	7, 9, 10	co 7409	. . . . . . 7 class (-∞(,)𝑎)
12	6, 11	cima 5680	. . . . . 6 class (◡𝑓 “ (-∞(,)𝑎))
13	2	cv 1541	. . . . . . 7 class 𝑠
14	5	cdm 5677	. . . . . . 7 class dom 𝑓
15		crest 17366	. . . . . . 7 class ↾t
16	13, 14, 15	co 7409	. . . . . 6 class (𝑠 ↾t dom 𝑓)
17	12, 16	wcel 2107	. . . . 5 wff (◡𝑓 “ (-∞(,)𝑎)) ∈ (𝑠 ↾t dom 𝑓)
18		cr 11109	. . . . 5 class ℝ
19	17, 8, 18	wral 3062	. . . 4 wff ∀𝑎 ∈ ℝ (◡𝑓 “ (-∞(,)𝑎)) ∈ (𝑠 ↾t dom 𝑓)
20	13	cuni 4909	. . . . 5 class ∪ 𝑠
21		cpm 8821	. . . . 5 class ↑pm
22	18, 20, 21	co 7409	. . . 4 class (ℝ ↑pm ∪ 𝑠)
23	19, 4, 22	crab 3433	. . 3 class {𝑓 ∈ (ℝ ↑pm ∪ 𝑠) ∣ ∀𝑎 ∈ ℝ (◡𝑓 “ (-∞(,)𝑎)) ∈ (𝑠 ↾t dom 𝑓)}
24	2, 3, 23	cmpt 5232	. 2 class (𝑠 ∈ SAlg ↦ {𝑓 ∈ (ℝ ↑pm ∪ 𝑠) ∣ ∀𝑎 ∈ ℝ (◡𝑓 “ (-∞(,)𝑎)) ∈ (𝑠 ↾t dom 𝑓)})
25	1, 24	wceq 1542	1 wff SMblFn = (𝑠 ∈ SAlg ↦ {𝑓 ∈ (ℝ ↑pm ∪ 𝑠) ∣ ∀𝑎 ∈ ℝ (◡𝑓 “ (-∞(,)𝑎)) ∈ (𝑠 ↾t dom 𝑓)})

This definition is referenced by:  issmflem  45443