Theorem issmfge 43754

Description: The predicate "𝐹 is a real-valued measurable function w.r.t. to the sigma-algebra 𝑆". A function is measurable iff the preimages of all left-closed intervals unbounded above are in the subspace sigma-algebra induced by its domain. The domain of 𝐹 is required to be b subset of the underlying set of 𝑆. Definition 121C of [Fremlin1] p. 36, and Proposition 121B (iv) of [Fremlin1] p. 36 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypotheses

Ref	Expression
issmfge.s	⊢ (𝜑 → 𝑆 ∈ SAlg)
issmfge.d	⊢ 𝐷 = dom 𝐹

Assertion

Ref	Expression
issmfge	⊢ (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ 𝑎 ≤ (𝐹‘𝑥)} ∈ (𝑆 ↾t 𝐷))))

Distinct variable groups:   𝐷,𝑎,𝑥   𝐹,𝑎,𝑥   𝑆,𝑎

Allowed substitution hints:   𝜑(𝑥,𝑎)   𝑆(𝑥)

Proof of Theorem issmfge

Dummy variables 𝑏 𝑦 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		issmfge.s	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ SAlg)
2	1	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → 𝑆 ∈ SAlg)
3		simpr 489	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → 𝐹 ∈ (SMblFn‘𝑆))
4		issmfge.d	. . . . . 6 ⊢ 𝐷 = dom 𝐹
5	2, 3, 4	smfdmss 43718	. . . . 5 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → 𝐷 ⊆ ∪ 𝑆)
6	2, 3, 4	smff 43717	. . . . 5 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → 𝐹:𝐷⟶ℝ)
7		nfv 1916	. . . . . . . . 9 ⊢ Ⅎ𝑦𝜑
8		nfv 1916	. . . . . . . . 9 ⊢ Ⅎ𝑦 𝐹 ∈ (SMblFn‘𝑆)
9	7, 8	nfan 1901	. . . . . . . 8 ⊢ Ⅎ𝑦(𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆))
10		nfv 1916	. . . . . . . 8 ⊢ Ⅎ𝑦 𝑏 ∈ ℝ
11	9, 10	nfan 1901	. . . . . . 7 ⊢ Ⅎ𝑦((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ)
12		nfv 1916	. . . . . . 7 ⊢ Ⅎ𝑐((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ)
13	1	uniexd 7459	. . . . . . . . . . . 12 ⊢ (𝜑 → ∪ 𝑆 ∈ V)
14	13	adantr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐷 ⊆ ∪ 𝑆) → ∪ 𝑆 ∈ V)
15		simpr 489	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐷 ⊆ ∪ 𝑆) → 𝐷 ⊆ ∪ 𝑆)
16	14, 15	ssexd 5187	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐷 ⊆ ∪ 𝑆) → 𝐷 ∈ V)
17	5, 16	syldan 595	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → 𝐷 ∈ V)
18		eqid 2759	. . . . . . . . 9 ⊢ (𝑆 ↾t 𝐷) = (𝑆 ↾t 𝐷)
19	2, 17, 18	subsalsal 43350	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → (𝑆 ↾t 𝐷) ∈ SAlg)
20	19	adantr 485	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) → (𝑆 ↾t 𝐷) ∈ SAlg)
21	6	ffvelrnda 6835	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑦 ∈ 𝐷) → (𝐹‘𝑦) ∈ ℝ)
22	21	rexrd 10714	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑦 ∈ 𝐷) → (𝐹‘𝑦) ∈ ℝ*)
23	22	adantlr 715	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) ∧ 𝑦 ∈ 𝐷) → (𝐹‘𝑦) ∈ ℝ*)
24	2	adantr 485	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑐 ∈ ℝ) → 𝑆 ∈ SAlg)
25	3	adantr 485	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑐 ∈ ℝ) → 𝐹 ∈ (SMblFn‘𝑆))
26		simpr 489	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑐 ∈ ℝ) → 𝑐 ∈ ℝ)
27	24, 25, 4, 26	smfpreimagt 43746	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑐 ∈ ℝ) → {𝑦 ∈ 𝐷 ∣ 𝑐 < (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷))
28	27	adantlr 715	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) ∧ 𝑐 ∈ ℝ) → {𝑦 ∈ 𝐷 ∣ 𝑐 < (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷))
29		simpr 489	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) → 𝑏 ∈ ℝ)
30	11, 12, 20, 23, 28, 29	salpreimagtge 43710	. . . . . 6 ⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) → {𝑦 ∈ 𝐷 ∣ 𝑏 ≤ (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷))
31	30	ralrimiva 3111	. . . . 5 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ 𝑏 ≤ (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷))
32	5, 6, 31	3jca 1126	. . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ 𝑏 ≤ (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷)))
33	32	ex 417	. . 3 ⊢ (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) → (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ 𝑏 ≤ (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷))))
34		nfv 1916	. . . . . . 7 ⊢ Ⅎ𝑦 𝐷 ⊆ ∪ 𝑆
35		nfv 1916	. . . . . . 7 ⊢ Ⅎ𝑦 𝐹:𝐷⟶ℝ
36		nfcv 2917	. . . . . . . 8 ⊢ Ⅎ𝑦ℝ
37		nfrab1 3300	. . . . . . . . 9 ⊢ Ⅎ𝑦{𝑦 ∈ 𝐷 ∣ 𝑏 ≤ (𝐹‘𝑦)}
38		nfcv 2917	. . . . . . . . 9 ⊢ Ⅎ𝑦(𝑆 ↾t 𝐷)
39	37, 38	nfel 2931	. . . . . . . 8 ⊢ Ⅎ𝑦{𝑦 ∈ 𝐷 ∣ 𝑏 ≤ (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷)
40	36, 39	nfralw 3151	. . . . . . 7 ⊢ Ⅎ𝑦∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ 𝑏 ≤ (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷)
41	34, 35, 40	nf3an 1903	. . . . . 6 ⊢ Ⅎ𝑦(𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ 𝑏 ≤ (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷))
42	7, 41	nfan 1901	. . . . 5 ⊢ Ⅎ𝑦(𝜑 ∧ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ 𝑏 ≤ (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷)))
43		nfv 1916	. . . . . 6 ⊢ Ⅎ𝑏𝜑
44		nfv 1916	. . . . . . 7 ⊢ Ⅎ𝑏 𝐷 ⊆ ∪ 𝑆
45		nfv 1916	. . . . . . 7 ⊢ Ⅎ𝑏 𝐹:𝐷⟶ℝ
46		nfra1 3145	. . . . . . 7 ⊢ Ⅎ𝑏∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ 𝑏 ≤ (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷)
47	44, 45, 46	nf3an 1903	. . . . . 6 ⊢ Ⅎ𝑏(𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ 𝑏 ≤ (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷))
48	43, 47	nfan 1901	. . . . 5 ⊢ Ⅎ𝑏(𝜑 ∧ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ 𝑏 ≤ (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷)))
49	1	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ 𝑏 ≤ (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷))) → 𝑆 ∈ SAlg)
50		simpr1 1192	. . . . 5 ⊢ ((𝜑 ∧ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ 𝑏 ≤ (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷))) → 𝐷 ⊆ ∪ 𝑆)
51		simpr2 1193	. . . . 5 ⊢ ((𝜑 ∧ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ 𝑏 ≤ (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷))) → 𝐹:𝐷⟶ℝ)
52		simpr3 1194	. . . . 5 ⊢ ((𝜑 ∧ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ 𝑏 ≤ (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷))) → ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ 𝑏 ≤ (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷))
53	42, 48, 49, 4, 50, 51, 52	issmfgelem 43753	. . . 4 ⊢ ((𝜑 ∧ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ 𝑏 ≤ (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷))) → 𝐹 ∈ (SMblFn‘𝑆))
54	53	ex 417	. . 3 ⊢ (𝜑 → ((𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ 𝑏 ≤ (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷)) → 𝐹 ∈ (SMblFn‘𝑆)))
55	33, 54	impbid 215	. 2 ⊢ (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ 𝑏 ≤ (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷))))
56		breq1 5028	. . . . . . . 8 ⊢ (𝑏 = 𝑎 → (𝑏 ≤ (𝐹‘𝑦) ↔ 𝑎 ≤ (𝐹‘𝑦)))
57	56	rabbidv 3390	. . . . . . 7 ⊢ (𝑏 = 𝑎 → {𝑦 ∈ 𝐷 ∣ 𝑏 ≤ (𝐹‘𝑦)} = {𝑦 ∈ 𝐷 ∣ 𝑎 ≤ (𝐹‘𝑦)})
58		fveq2 6651	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → (𝐹‘𝑦) = (𝐹‘𝑥))
59	58	breq2d 5037	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → (𝑎 ≤ (𝐹‘𝑦) ↔ 𝑎 ≤ (𝐹‘𝑥)))
60	59	cbvrabv 3402	. . . . . . . 8 ⊢ {𝑦 ∈ 𝐷 ∣ 𝑎 ≤ (𝐹‘𝑦)} = {𝑥 ∈ 𝐷 ∣ 𝑎 ≤ (𝐹‘𝑥)}
61	60	a1i 11	. . . . . . 7 ⊢ (𝑏 = 𝑎 → {𝑦 ∈ 𝐷 ∣ 𝑎 ≤ (𝐹‘𝑦)} = {𝑥 ∈ 𝐷 ∣ 𝑎 ≤ (𝐹‘𝑥)})
62	57, 61	eqtrd 2794	. . . . . 6 ⊢ (𝑏 = 𝑎 → {𝑦 ∈ 𝐷 ∣ 𝑏 ≤ (𝐹‘𝑦)} = {𝑥 ∈ 𝐷 ∣ 𝑎 ≤ (𝐹‘𝑥)})
63	62	eleq1d 2835	. . . . 5 ⊢ (𝑏 = 𝑎 → ({𝑦 ∈ 𝐷 ∣ 𝑏 ≤ (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷) ↔ {𝑥 ∈ 𝐷 ∣ 𝑎 ≤ (𝐹‘𝑥)} ∈ (𝑆 ↾t 𝐷)))
64	63	cbvralvw 3359	. . . 4 ⊢ (∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ 𝑏 ≤ (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷) ↔ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ 𝑎 ≤ (𝐹‘𝑥)} ∈ (𝑆 ↾t 𝐷))
65	64	3anbi3i 1157	. . 3 ⊢ ((𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ 𝑏 ≤ (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷)) ↔ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ 𝑎 ≤ (𝐹‘𝑥)} ∈ (𝑆 ↾t 𝐷)))
66	65	a1i 11	. 2 ⊢ (𝜑 → ((𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ 𝑏 ≤ (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷)) ↔ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ 𝑎 ≤ (𝐹‘𝑥)} ∈ (𝑆 ↾t 𝐷))))
67	55, 66	bitrd 282	1 ⊢ (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ 𝑎 ≤ (𝐹‘𝑥)} ∈ (𝑆 ↾t 𝐷))))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1085   = wceq 1539   ∈ wcel 2112  ∀wral 3068  {crab 3072  Vcvv 3407   ⊆ wss 3854  ∪ cuni 4791   class class class wbr 5025  dom cdm 5517  ⟶wf 6324  ‘cfv 6328  (class class class)co 7143  ℝcr 10559  ℝ^*cxr 10697   < clt 10698   ≤ cle 10699   ↾_t crest 16737  SAlgcsalg 43301  SMblFncsmblfn 43685

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-rep 5149  ax-sep 5162  ax-nul 5169  ax-pow 5227  ax-pr 5291  ax-un 7452  ax-inf2 9122  ax-cc 9880  ax-ac2 9908  ax-cnex 10616  ax-resscn 10617  ax-1cn 10618  ax-icn 10619  ax-addcl 10620  ax-addrcl 10621  ax-mulcl 10622  ax-mulrcl 10623  ax-mulcom 10624  ax-addass 10625  ax-mulass 10626  ax-distr 10627  ax-i2m1 10628  ax-1ne0 10629  ax-1rid 10630  ax-rnegex 10631  ax-rrecex 10632  ax-cnre 10633  ax-pre-lttri 10634  ax-pre-lttrn 10635  ax-pre-ltadd 10636  ax-pre-mulgt0 10637  ax-pre-sup 10638

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2899  df-ne 2950  df-nel 3054  df-ral 3073  df-rex 3074  df-reu 3075  df-rmo 3076  df-rab 3077  df-v 3409  df-sbc 3694  df-csb 3802  df-dif 3857  df-un 3859  df-in 3861  df-ss 3871  df-pss 3873  df-nul 4222  df-if 4414  df-pw 4489  df-sn 4516  df-pr 4518  df-tp 4520  df-op 4522  df-uni 4792  df-int 4832  df-iun 4878  df-iin 4879  df-br 5026  df-opab 5088  df-mpt 5106  df-tr 5132  df-id 5423  df-eprel 5428  df-po 5436  df-so 5437  df-fr 5476  df-se 5477  df-we 5478  df-xp 5523  df-rel 5524  df-cnv 5525  df-co 5526  df-dm 5527  df-rn 5528  df-res 5529  df-ima 5530  df-pred 6119  df-ord 6165  df-on 6166  df-lim 6167  df-suc 6168  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-isom 6337  df-riota 7101  df-ov 7146  df-oprab 7147  df-mpo 7148  df-om 7573  df-1st 7686  df-2nd 7687  df-wrecs 7950  df-recs 8011  df-rdg 8049  df-1o 8105  df-oadd 8109  df-er 8292  df-map 8411  df-pm 8412  df-en 8521  df-dom 8522  df-sdom 8523  df-fin 8524  df-sup 8924  df-inf 8925  df-card 9386  df-acn 9389  df-ac 9561  df-pnf 10700  df-mnf 10701  df-xr 10702  df-ltxr 10703  df-le 10704  df-sub 10895  df-neg 10896  df-div 11321  df-nn 11660  df-n0 11920  df-z 12006  df-uz 12268  df-q 12374  df-rp 12416  df-ioo 12768  df-ico 12770  df-fl 13196  df-rest 16739  df-salg 43302  df-smblfn 43686

This theorem is referenced by:  smfpreimage  43766