Theorem issmflem 47089

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 open intervals unbounded below are in the subspace sigma-algebra induced by its domain. The domain of 𝐹 is required to be a subset of the underlying set of 𝑆. Definition 121C of [Fremlin1] p. 36, and Proposition 121B (i) of [Fremlin1] p. 35 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypotheses

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

Assertion

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

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

Proof of Theorem issmflem

Dummy variables 𝑓 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → 𝐹 ∈ (SMblFn‘𝑆))
2		df-smblfn 47058	. . . . . . . . 9 ⊢ SMblFn = (𝑠 ∈ SAlg ↦ {𝑓 ∈ (ℝ ↑pm ∪ 𝑠) ∣ ∀𝑎 ∈ ℝ (◡𝑓 “ (-∞(,)𝑎)) ∈ (𝑠 ↾t dom 𝑓)})
3		unieq 4876	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑆 → ∪ 𝑠 = ∪ 𝑆)
4	3	oveq2d 7384	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑆 → (ℝ ↑pm ∪ 𝑠) = (ℝ ↑pm ∪ 𝑆))
5	4	rabeqdv 3416	. . . . . . . . . 10 ⊢ (𝑠 = 𝑆 → {𝑓 ∈ (ℝ ↑pm ∪ 𝑠) ∣ ∀𝑎 ∈ ℝ (◡𝑓 “ (-∞(,)𝑎)) ∈ (𝑠 ↾t dom 𝑓)} = {𝑓 ∈ (ℝ ↑pm ∪ 𝑆) ∣ ∀𝑎 ∈ ℝ (◡𝑓 “ (-∞(,)𝑎)) ∈ (𝑠 ↾t dom 𝑓)})
6		oveq1 7375	. . . . . . . . . . . . 13 ⊢ (𝑠 = 𝑆 → (𝑠 ↾t dom 𝑓) = (𝑆 ↾t dom 𝑓))
7	6	eleq2d 2823	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑆 → ((◡𝑓 “ (-∞(,)𝑎)) ∈ (𝑠 ↾t dom 𝑓) ↔ (◡𝑓 “ (-∞(,)𝑎)) ∈ (𝑆 ↾t dom 𝑓)))
8	7	ralbidv 3161	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑆 → (∀𝑎 ∈ ℝ (◡𝑓 “ (-∞(,)𝑎)) ∈ (𝑠 ↾t dom 𝑓) ↔ ∀𝑎 ∈ ℝ (◡𝑓 “ (-∞(,)𝑎)) ∈ (𝑆 ↾t dom 𝑓)))
9	8	rabbidv 3408	. . . . . . . . . 10 ⊢ (𝑠 = 𝑆 → {𝑓 ∈ (ℝ ↑pm ∪ 𝑆) ∣ ∀𝑎 ∈ ℝ (◡𝑓 “ (-∞(,)𝑎)) ∈ (𝑠 ↾t dom 𝑓)} = {𝑓 ∈ (ℝ ↑pm ∪ 𝑆) ∣ ∀𝑎 ∈ ℝ (◡𝑓 “ (-∞(,)𝑎)) ∈ (𝑆 ↾t dom 𝑓)})
10	5, 9	eqtrd 2772	. . . . . . . . 9 ⊢ (𝑠 = 𝑆 → {𝑓 ∈ (ℝ ↑pm ∪ 𝑠) ∣ ∀𝑎 ∈ ℝ (◡𝑓 “ (-∞(,)𝑎)) ∈ (𝑠 ↾t dom 𝑓)} = {𝑓 ∈ (ℝ ↑pm ∪ 𝑆) ∣ ∀𝑎 ∈ ℝ (◡𝑓 “ (-∞(,)𝑎)) ∈ (𝑆 ↾t dom 𝑓)})
11		issmflem.s	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ SAlg)
12		ovex 7401	. . . . . . . . . . 11 ⊢ (ℝ ↑pm ∪ 𝑆) ∈ V
13	12	rabex 5286	. . . . . . . . . 10 ⊢ {𝑓 ∈ (ℝ ↑pm ∪ 𝑆) ∣ ∀𝑎 ∈ ℝ (◡𝑓 “ (-∞(,)𝑎)) ∈ (𝑆 ↾t dom 𝑓)} ∈ V
14	13	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → {𝑓 ∈ (ℝ ↑pm ∪ 𝑆) ∣ ∀𝑎 ∈ ℝ (◡𝑓 “ (-∞(,)𝑎)) ∈ (𝑆 ↾t dom 𝑓)} ∈ V)
15	2, 10, 11, 14	fvmptd3 6973	. . . . . . . 8 ⊢ (𝜑 → (SMblFn‘𝑆) = {𝑓 ∈ (ℝ ↑pm ∪ 𝑆) ∣ ∀𝑎 ∈ ℝ (◡𝑓 “ (-∞(,)𝑎)) ∈ (𝑆 ↾t dom 𝑓)})
16	15	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → (SMblFn‘𝑆) = {𝑓 ∈ (ℝ ↑pm ∪ 𝑆) ∣ ∀𝑎 ∈ ℝ (◡𝑓 “ (-∞(,)𝑎)) ∈ (𝑆 ↾t dom 𝑓)})
17	1, 16	eleqtrd 2839	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → 𝐹 ∈ {𝑓 ∈ (ℝ ↑pm ∪ 𝑆) ∣ ∀𝑎 ∈ ℝ (◡𝑓 “ (-∞(,)𝑎)) ∈ (𝑆 ↾t dom 𝑓)})
18		elrabi 3644	. . . . . 6 ⊢ (𝐹 ∈ {𝑓 ∈ (ℝ ↑pm ∪ 𝑆) ∣ ∀𝑎 ∈ ℝ (◡𝑓 “ (-∞(,)𝑎)) ∈ (𝑆 ↾t dom 𝑓)} → 𝐹 ∈ (ℝ ↑pm ∪ 𝑆))
19	17, 18	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → 𝐹 ∈ (ℝ ↑pm ∪ 𝑆))
20		issmflem.d	. . . . . . 7 ⊢ 𝐷 = dom 𝐹
21		elpmi2 45587	. . . . . . 7 ⊢ (𝐹 ∈ (ℝ ↑pm ∪ 𝑆) → dom 𝐹 ⊆ ∪ 𝑆)
22	20, 21	eqsstrid 3974	. . . . . 6 ⊢ (𝐹 ∈ (ℝ ↑pm ∪ 𝑆) → 𝐷 ⊆ ∪ 𝑆)
23	22	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝐹 ∈ (ℝ ↑pm ∪ 𝑆)) → 𝐷 ⊆ ∪ 𝑆)
24	19, 23	syldan 592	. . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → 𝐷 ⊆ ∪ 𝑆)
25		elpmi 8795	. . . . . . 7 ⊢ (𝐹 ∈ (ℝ ↑pm ∪ 𝑆) → (𝐹:dom 𝐹⟶ℝ ∧ dom 𝐹 ⊆ ∪ 𝑆))
26	19, 25	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → (𝐹:dom 𝐹⟶ℝ ∧ dom 𝐹 ⊆ ∪ 𝑆))
27	26	simpld 494	. . . . 5 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → 𝐹:dom 𝐹⟶ℝ)
28	20	feq2i 6662	. . . . . 6 ⊢ (𝐹:𝐷⟶ℝ ↔ 𝐹:dom 𝐹⟶ℝ)
29	28	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → (𝐹:𝐷⟶ℝ ↔ 𝐹:dom 𝐹⟶ℝ))
30	27, 29	mpbird 257	. . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → 𝐹:𝐷⟶ℝ)
31		cnveq 5830	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝐹 → ◡𝑓 = ◡𝐹)
32	31	imaeq1d 6026	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝐹 → (◡𝑓 “ (-∞(,)𝑎)) = (◡𝐹 “ (-∞(,)𝑎)))
33		dmeq 5860	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
34	33	oveq2d 7384	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝐹 → (𝑆 ↾t dom 𝑓) = (𝑆 ↾t dom 𝐹))
35	32, 34	eleq12d 2831	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝐹 → ((◡𝑓 “ (-∞(,)𝑎)) ∈ (𝑆 ↾t dom 𝑓) ↔ (◡𝐹 “ (-∞(,)𝑎)) ∈ (𝑆 ↾t dom 𝐹)))
36	35	ralbidv 3161	. . . . . . . . . . 11 ⊢ (𝑓 = 𝐹 → (∀𝑎 ∈ ℝ (◡𝑓 “ (-∞(,)𝑎)) ∈ (𝑆 ↾t dom 𝑓) ↔ ∀𝑎 ∈ ℝ (◡𝐹 “ (-∞(,)𝑎)) ∈ (𝑆 ↾t dom 𝐹)))
37	36	elrab 3648	. . . . . . . . . 10 ⊢ (𝐹 ∈ {𝑓 ∈ (ℝ ↑pm ∪ 𝑆) ∣ ∀𝑎 ∈ ℝ (◡𝑓 “ (-∞(,)𝑎)) ∈ (𝑆 ↾t dom 𝑓)} ↔ (𝐹 ∈ (ℝ ↑pm ∪ 𝑆) ∧ ∀𝑎 ∈ ℝ (◡𝐹 “ (-∞(,)𝑎)) ∈ (𝑆 ↾t dom 𝐹)))
38	37	simprbi 497	. . . . . . . . 9 ⊢ (𝐹 ∈ {𝑓 ∈ (ℝ ↑pm ∪ 𝑆) ∣ ∀𝑎 ∈ ℝ (◡𝑓 “ (-∞(,)𝑎)) ∈ (𝑆 ↾t dom 𝑓)} → ∀𝑎 ∈ ℝ (◡𝐹 “ (-∞(,)𝑎)) ∈ (𝑆 ↾t dom 𝐹))
39	17, 38	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → ∀𝑎 ∈ ℝ (◡𝐹 “ (-∞(,)𝑎)) ∈ (𝑆 ↾t dom 𝐹))
40	39	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑎 ∈ ℝ) → ∀𝑎 ∈ ℝ (◡𝐹 “ (-∞(,)𝑎)) ∈ (𝑆 ↾t dom 𝐹))
41		simpr 484	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑎 ∈ ℝ) → 𝑎 ∈ ℝ)
42		rspa 3227	. . . . . . 7 ⊢ ((∀𝑎 ∈ ℝ (◡𝐹 “ (-∞(,)𝑎)) ∈ (𝑆 ↾t dom 𝐹) ∧ 𝑎 ∈ ℝ) → (◡𝐹 “ (-∞(,)𝑎)) ∈ (𝑆 ↾t dom 𝐹))
43	40, 41, 42	syl2anc 585	. . . . . 6 ⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑎 ∈ ℝ) → (◡𝐹 “ (-∞(,)𝑎)) ∈ (𝑆 ↾t dom 𝐹))
44	30	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑎 ∈ ℝ) → 𝐹:𝐷⟶ℝ)
45		simpl 482	. . . . . . . . . 10 ⊢ ((𝐹:𝐷⟶ℝ ∧ 𝑎 ∈ ℝ) → 𝐹:𝐷⟶ℝ)
46		simpr 484	. . . . . . . . . . 11 ⊢ ((𝐹:𝐷⟶ℝ ∧ 𝑎 ∈ ℝ) → 𝑎 ∈ ℝ)
47	46	rexrd 11194	. . . . . . . . . 10 ⊢ ((𝐹:𝐷⟶ℝ ∧ 𝑎 ∈ ℝ) → 𝑎 ∈ ℝ*)
48	45, 47	preimaioomnf 47081	. . . . . . . . 9 ⊢ ((𝐹:𝐷⟶ℝ ∧ 𝑎 ∈ ℝ) → (◡𝐹 “ (-∞(,)𝑎)) = {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎})
49	48	eqcomd 2743	. . . . . . . 8 ⊢ ((𝐹:𝐷⟶ℝ ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} = (◡𝐹 “ (-∞(,)𝑎)))
50	44, 41, 49	syl2anc 585	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} = (◡𝐹 “ (-∞(,)𝑎)))
51	20	oveq2i 7379	. . . . . . . 8 ⊢ (𝑆 ↾t 𝐷) = (𝑆 ↾t dom 𝐹)
52	51	a1i 11	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑎 ∈ ℝ) → (𝑆 ↾t 𝐷) = (𝑆 ↾t dom 𝐹))
53	50, 52	eleq12d 2831	. . . . . 6 ⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑎 ∈ ℝ) → ({𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷) ↔ (◡𝐹 “ (-∞(,)𝑎)) ∈ (𝑆 ↾t dom 𝐹)))
54	43, 53	mpbird 257	. . . . 5 ⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷))
55	54	ralrimiva 3130	. . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷))
56	24, 30, 55	3jca 1129	. . 3 ⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷)))
57	56	ex 412	. 2 ⊢ (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) → (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷))))
58		reex 11129	. . . . . . . . 9 ⊢ ℝ ∈ V
59	58	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ)) → ℝ ∈ V)
60	11	uniexd 7697	. . . . . . . . 9 ⊢ (𝜑 → ∪ 𝑆 ∈ V)
61	60	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ)) → ∪ 𝑆 ∈ V)
62		simprr 773	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ)) → 𝐹:𝐷⟶ℝ)
63		fssxp 6697	. . . . . . . . . . . 12 ⊢ (𝐹:𝐷⟶ℝ → 𝐹 ⊆ (𝐷 × ℝ))
64	63	adantl 481	. . . . . . . . . . 11 ⊢ ((𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ) → 𝐹 ⊆ (𝐷 × ℝ))
65		xpss1 5651	. . . . . . . . . . . 12 ⊢ (𝐷 ⊆ ∪ 𝑆 → (𝐷 × ℝ) ⊆ (∪ 𝑆 × ℝ))
66	65	adantr 480	. . . . . . . . . . 11 ⊢ ((𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ) → (𝐷 × ℝ) ⊆ (∪ 𝑆 × ℝ))
67	64, 66	sstrd 3946	. . . . . . . . . 10 ⊢ ((𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ) → 𝐹 ⊆ (∪ 𝑆 × ℝ))
68	67	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ)) → 𝐹 ⊆ (∪ 𝑆 × ℝ))
69		dmss 5859	. . . . . . . . . . . 12 ⊢ (𝐹 ⊆ (∪ 𝑆 × ℝ) → dom 𝐹 ⊆ dom (∪ 𝑆 × ℝ))
70		dmxpss 6137	. . . . . . . . . . . . 13 ⊢ dom (∪ 𝑆 × ℝ) ⊆ ∪ 𝑆
71	70	a1i 11	. . . . . . . . . . . 12 ⊢ (𝐹 ⊆ (∪ 𝑆 × ℝ) → dom (∪ 𝑆 × ℝ) ⊆ ∪ 𝑆)
72	69, 71	sstrd 3946	. . . . . . . . . . 11 ⊢ (𝐹 ⊆ (∪ 𝑆 × ℝ) → dom 𝐹 ⊆ ∪ 𝑆)
73	72	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹 ⊆ (∪ 𝑆 × ℝ)) → dom 𝐹 ⊆ ∪ 𝑆)
74	20, 73	eqsstrid 3974	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐹 ⊆ (∪ 𝑆 × ℝ)) → 𝐷 ⊆ ∪ 𝑆)
75	68, 74	syldan 592	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ)) → 𝐷 ⊆ ∪ 𝑆)
76		elpm2r 8794	. . . . . . . 8 ⊢ (((ℝ ∈ V ∧ ∪ 𝑆 ∈ V) ∧ (𝐹:𝐷⟶ℝ ∧ 𝐷 ⊆ ∪ 𝑆)) → 𝐹 ∈ (ℝ ↑pm ∪ 𝑆))
77	59, 61, 62, 75, 76	syl22anc 839	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ)) → 𝐹 ∈ (ℝ ↑pm ∪ 𝑆))
78	77	3adantr3 1173	. . . . . 6 ⊢ ((𝜑 ∧ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷))) → 𝐹 ∈ (ℝ ↑pm ∪ 𝑆))
79	20	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝐹:𝐷⟶ℝ ∧ 𝑎 ∈ ℝ) → 𝐷 = dom 𝐹)
80	79	oveq2d 7384	. . . . . . . . . . . 12 ⊢ ((𝐹:𝐷⟶ℝ ∧ 𝑎 ∈ ℝ) → (𝑆 ↾t 𝐷) = (𝑆 ↾t dom 𝐹))
81	49, 80	eleq12d 2831	. . . . . . . . . . 11 ⊢ ((𝐹:𝐷⟶ℝ ∧ 𝑎 ∈ ℝ) → ({𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷) ↔ (◡𝐹 “ (-∞(,)𝑎)) ∈ (𝑆 ↾t dom 𝐹)))
82	81	ralbidva 3159	. . . . . . . . . 10 ⊢ (𝐹:𝐷⟶ℝ → (∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷) ↔ ∀𝑎 ∈ ℝ (◡𝐹 “ (-∞(,)𝑎)) ∈ (𝑆 ↾t dom 𝐹)))
83	82	biimpd 229	. . . . . . . . 9 ⊢ (𝐹:𝐷⟶ℝ → (∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷) → ∀𝑎 ∈ ℝ (◡𝐹 “ (-∞(,)𝑎)) ∈ (𝑆 ↾t dom 𝐹)))
84	83	imp 406	. . . . . . . 8 ⊢ ((𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷)) → ∀𝑎 ∈ ℝ (◡𝐹 “ (-∞(,)𝑎)) ∈ (𝑆 ↾t dom 𝐹))
85	84	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷))) → ∀𝑎 ∈ ℝ (◡𝐹 “ (-∞(,)𝑎)) ∈ (𝑆 ↾t dom 𝐹))
86	85	3adantr1 1171	. . . . . 6 ⊢ ((𝜑 ∧ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷))) → ∀𝑎 ∈ ℝ (◡𝐹 “ (-∞(,)𝑎)) ∈ (𝑆 ↾t dom 𝐹))
87	78, 86	jca 511	. . . . 5 ⊢ ((𝜑 ∧ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷))) → (𝐹 ∈ (ℝ ↑pm ∪ 𝑆) ∧ ∀𝑎 ∈ ℝ (◡𝐹 “ (-∞(,)𝑎)) ∈ (𝑆 ↾t dom 𝐹)))
88	87, 37	sylibr 234	. . . 4 ⊢ ((𝜑 ∧ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷))) → 𝐹 ∈ {𝑓 ∈ (ℝ ↑pm ∪ 𝑆) ∣ ∀𝑎 ∈ ℝ (◡𝑓 “ (-∞(,)𝑎)) ∈ (𝑆 ↾t dom 𝑓)})
89	15	eqcomd 2743	. . . . 5 ⊢ (𝜑 → {𝑓 ∈ (ℝ ↑pm ∪ 𝑆) ∣ ∀𝑎 ∈ ℝ (◡𝑓 “ (-∞(,)𝑎)) ∈ (𝑆 ↾t dom 𝑓)} = (SMblFn‘𝑆))
90	89	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷))) → {𝑓 ∈ (ℝ ↑pm ∪ 𝑆) ∣ ∀𝑎 ∈ ℝ (◡𝑓 “ (-∞(,)𝑎)) ∈ (𝑆 ↾t dom 𝑓)} = (SMblFn‘𝑆))
91	88, 90	eleqtrd 2839	. . 3 ⊢ ((𝜑 ∧ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷))) → 𝐹 ∈ (SMblFn‘𝑆))
92	91	ex 412	. 2 ⊢ (𝜑 → ((𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷)) → 𝐹 ∈ (SMblFn‘𝑆)))
93	57, 92	impbid 212	1 ⊢ (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052  {crab 3401  Vcvv 3442   ⊆ wss 3903  ∪ cuni 4865   class class class wbr 5100   × cxp 5630  ^◡ccnv 5631  dom cdm 5632   “ cima 5635  ⟶wf 6496  ‘cfv 6500  (class class class)co 7368   ↑_pm cpm 8776  ℝcr 11037  -∞cmnf 11176   < clt 11178  (,)cioo 13273   ↾_t crest 17352  SAlgcsalg 46670  SMblFncsmblfn 47057

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-cnex 11094  ax-resscn 11095  ax-pre-lttri 11112  ax-pre-lttrn 11113

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-po 5540  df-so 5541  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-1st 7943  df-2nd 7944  df-er 8645  df-pm 8778  df-en 8896  df-dom 8897  df-sdom 8898  df-pnf 11180  df-mnf 11181  df-xr 11182  df-ltxr 11183  df-le 11184  df-ioo 13277  df-ico 13279  df-smblfn 47058

This theorem is referenced by:  issmf  47090