Theorem issmfge 46757

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 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → 𝑆 ∈ SAlg)
3		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → 𝐹 ∈ (SMblFn‘𝑆))
4		issmfge.d	. . . . . 6 ⊢ 𝐷 = dom 𝐹
5	2, 3, 4	smfdmss 46720	. . . . 5 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → 𝐷 ⊆ ∪ 𝑆)
6	2, 3, 4	smff 46719	. . . . 5 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → 𝐹:𝐷⟶ℝ)
7		nfv 1913	. . . . . . . . 9 ⊢ Ⅎ𝑦𝜑
8		nfv 1913	. . . . . . . . 9 ⊢ Ⅎ𝑦 𝐹 ∈ (SMblFn‘𝑆)
9	7, 8	nfan 1898	. . . . . . . 8 ⊢ Ⅎ𝑦(𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆))
10		nfv 1913	. . . . . . . 8 ⊢ Ⅎ𝑦 𝑏 ∈ ℝ
11	9, 10	nfan 1898	. . . . . . 7 ⊢ Ⅎ𝑦((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ)
12		nfv 1913	. . . . . . 7 ⊢ Ⅎ𝑐((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ)
13	1	uniexd 7744	. . . . . . . . . . . 12 ⊢ (𝜑 → ∪ 𝑆 ∈ V)
14	13	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐷 ⊆ ∪ 𝑆) → ∪ 𝑆 ∈ V)
15		simpr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐷 ⊆ ∪ 𝑆) → 𝐷 ⊆ ∪ 𝑆)
16	14, 15	ssexd 5304	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐷 ⊆ ∪ 𝑆) → 𝐷 ∈ V)
17	5, 16	syldan 591	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → 𝐷 ∈ V)
18		eqid 2734	. . . . . . . . 9 ⊢ (𝑆 ↾t 𝐷) = (𝑆 ↾t 𝐷)
19	2, 17, 18	subsalsal 46346	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → (𝑆 ↾t 𝐷) ∈ SAlg)
20	19	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) → (𝑆 ↾t 𝐷) ∈ SAlg)
21	6	ffvelcdmda 7084	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑦 ∈ 𝐷) → (𝐹‘𝑦) ∈ ℝ)
22	21	rexrd 11293	. . . . . . . 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 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑐 ∈ ℝ) → 𝑆 ∈ SAlg)
25	3	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑐 ∈ ℝ) → 𝐹 ∈ (SMblFn‘𝑆))
26		simpr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑐 ∈ ℝ) → 𝑐 ∈ ℝ)
27	24, 25, 4, 26	smfpreimagt 46749	. . . . . . . 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 484	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) → 𝑏 ∈ ℝ)
30	11, 12, 20, 23, 28, 29	salpreimagtge 46712	. . . . . 6 ⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) → {𝑦 ∈ 𝐷 ∣ 𝑏 ≤ (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷))
31	30	ralrimiva 3133	. . . . 5 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ 𝑏 ≤ (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷))
32	5, 6, 31	3jca 1128	. . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ 𝑏 ≤ (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷)))
33	32	ex 412	. . 3 ⊢ (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) → (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ 𝑏 ≤ (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷))))
34		nfv 1913	. . . . . . 7 ⊢ Ⅎ𝑦 𝐷 ⊆ ∪ 𝑆
35		nfv 1913	. . . . . . 7 ⊢ Ⅎ𝑦 𝐹:𝐷⟶ℝ
36		nfcv 2897	. . . . . . . 8 ⊢ Ⅎ𝑦ℝ
37		nfrab1 3440	. . . . . . . . 9 ⊢ Ⅎ𝑦{𝑦 ∈ 𝐷 ∣ 𝑏 ≤ (𝐹‘𝑦)}
38		nfcv 2897	. . . . . . . . 9 ⊢ Ⅎ𝑦(𝑆 ↾t 𝐷)
39	37, 38	nfel 2912	. . . . . . . 8 ⊢ Ⅎ𝑦{𝑦 ∈ 𝐷 ∣ 𝑏 ≤ (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷)
40	36, 39	nfralw 3294	. . . . . . 7 ⊢ Ⅎ𝑦∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ 𝑏 ≤ (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷)
41	34, 35, 40	nf3an 1900	. . . . . 6 ⊢ Ⅎ𝑦(𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ 𝑏 ≤ (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷))
42	7, 41	nfan 1898	. . . . 5 ⊢ Ⅎ𝑦(𝜑 ∧ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ 𝑏 ≤ (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷)))
43		nfv 1913	. . . . . 6 ⊢ Ⅎ𝑏𝜑
44		nfv 1913	. . . . . . 7 ⊢ Ⅎ𝑏 𝐷 ⊆ ∪ 𝑆
45		nfv 1913	. . . . . . 7 ⊢ Ⅎ𝑏 𝐹:𝐷⟶ℝ
46		nfra1 3269	. . . . . . 7 ⊢ Ⅎ𝑏∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ 𝑏 ≤ (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷)
47	44, 45, 46	nf3an 1900	. . . . . 6 ⊢ Ⅎ𝑏(𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ 𝑏 ≤ (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷))
48	43, 47	nfan 1898	. . . . 5 ⊢ Ⅎ𝑏(𝜑 ∧ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ 𝑏 ≤ (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷)))
49	1	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ 𝑏 ≤ (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷))) → 𝑆 ∈ SAlg)
50		simpr1 1194	. . . . 5 ⊢ ((𝜑 ∧ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ 𝑏 ≤ (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷))) → 𝐷 ⊆ ∪ 𝑆)
51		simpr2 1195	. . . . 5 ⊢ ((𝜑 ∧ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ 𝑏 ≤ (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷))) → 𝐹:𝐷⟶ℝ)
52		simpr3 1196	. . . . 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 46756	. . . 4 ⊢ ((𝜑 ∧ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ 𝑏 ≤ (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷))) → 𝐹 ∈ (SMblFn‘𝑆))
54	53	ex 412	. . 3 ⊢ (𝜑 → ((𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ 𝑏 ≤ (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷)) → 𝐹 ∈ (SMblFn‘𝑆)))
55	33, 54	impbid 212	. 2 ⊢ (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ 𝑏 ≤ (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷))))
56		breq1 5126	. . . . . . . 8 ⊢ (𝑏 = 𝑎 → (𝑏 ≤ (𝐹‘𝑦) ↔ 𝑎 ≤ (𝐹‘𝑦)))
57	56	rabbidv 3427	. . . . . . 7 ⊢ (𝑏 = 𝑎 → {𝑦 ∈ 𝐷 ∣ 𝑏 ≤ (𝐹‘𝑦)} = {𝑦 ∈ 𝐷 ∣ 𝑎 ≤ (𝐹‘𝑦)})
58		fveq2 6886	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → (𝐹‘𝑦) = (𝐹‘𝑥))
59	58	breq2d 5135	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → (𝑎 ≤ (𝐹‘𝑦) ↔ 𝑎 ≤ (𝐹‘𝑥)))
60	59	cbvrabv 3430	. . . . . . . 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 2769	. . . . . 6 ⊢ (𝑏 = 𝑎 → {𝑦 ∈ 𝐷 ∣ 𝑏 ≤ (𝐹‘𝑦)} = {𝑥 ∈ 𝐷 ∣ 𝑎 ≤ (𝐹‘𝑥)})
63	62	eleq1d 2818	. . . . 5 ⊢ (𝑏 = 𝑎 → ({𝑦 ∈ 𝐷 ∣ 𝑏 ≤ (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷) ↔ {𝑥 ∈ 𝐷 ∣ 𝑎 ≤ (𝐹‘𝑥)} ∈ (𝑆 ↾t 𝐷)))
64	63	cbvralvw 3223	. . . 4 ⊢ (∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ 𝑏 ≤ (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷) ↔ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ 𝑎 ≤ (𝐹‘𝑥)} ∈ (𝑆 ↾t 𝐷))
65	64	3anbi3i 1159	. . 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 279	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 206   ∧ wa 395   ∧ w3a 1086   = wceq 1539   ∈ wcel 2107  ∀wral 3050  {crab 3419  Vcvv 3463   ⊆ wss 3931  ∪ cuni 4887   class class class wbr 5123  dom cdm 5665  ⟶wf 6537  ‘cfv 6541  (class class class)co 7413  ℝcr 11136  ℝ^*cxr 11276   < clt 11277   ≤ cle 11278   ↾_t crest 17437  SAlgcsalg 46295  SMblFncsmblfn 46682

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-rep 5259  ax-sep 5276  ax-nul 5286  ax-pow 5345  ax-pr 5412  ax-un 7737  ax-inf2 9663  ax-cc 10457  ax-ac2 10485  ax-cnex 11193  ax-resscn 11194  ax-1cn 11195  ax-icn 11196  ax-addcl 11197  ax-addrcl 11198  ax-mulcl 11199  ax-mulrcl 11200  ax-mulcom 11201  ax-addass 11202  ax-mulass 11203  ax-distr 11204  ax-i2m1 11205  ax-1ne0 11206  ax-1rid 11207  ax-rnegex 11208  ax-rrecex 11209  ax-cnre 11210  ax-pre-lttri 11211  ax-pre-lttrn 11212  ax-pre-ltadd 11213  ax-pre-mulgt0 11214  ax-pre-sup 11215

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-int 4927  df-iun 4973  df-iin 4974  df-br 5124  df-opab 5186  df-mpt 5206  df-tr 5240  df-id 5558  df-eprel 5564  df-po 5572  df-so 5573  df-fr 5617  df-se 5618  df-we 5619  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-pred 6301  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-isom 6550  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7870  df-1st 7996  df-2nd 7997  df-frecs 8288  df-wrecs 8319  df-recs 8393  df-rdg 8432  df-1o 8488  df-er 8727  df-map 8850  df-pm 8851  df-en 8968  df-dom 8969  df-sdom 8970  df-fin 8971  df-sup 9464  df-inf 9465  df-card 9961  df-acn 9964  df-ac 10138  df-pnf 11279  df-mnf 11280  df-xr 11281  df-ltxr 11282  df-le 11283  df-sub 11476  df-neg 11477  df-div 11903  df-nn 12249  df-n0 12510  df-z 12597  df-uz 12861  df-q 12973  df-rp 13017  df-ioo 13373  df-ico 13375  df-fl 13814  df-rest 17439  df-salg 46296  df-smblfn 46683

This theorem is referenced by:  smfpreimage  46769