Theorem issmff 45748

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
issmff.x	⊢ Ⅎ𝑥𝐹
issmff.s	⊢ (𝜑 → 𝑆 ∈ SAlg)
issmff.d	⊢ 𝐷 = dom 𝐹

Assertion

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

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

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

Proof of Theorem issmff

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		issmff.s	. . 3 ⊢ (𝜑 → 𝑆 ∈ SAlg)
2		issmff.d	. . 3 ⊢ 𝐷 = dom 𝐹
3	1, 2	issmf 45742	. 2 ⊢ (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) < 𝑎} ∈ (𝑆 ↾t 𝐷))))
4		nfcv 2901	. . . . . . 7 ⊢ Ⅎ𝑦𝐷
5		issmff.x	. . . . . . . . 9 ⊢ Ⅎ𝑥𝐹
6	5	nfdm 5949	. . . . . . . 8 ⊢ Ⅎ𝑥dom 𝐹
7	2, 6	nfcxfr 2899	. . . . . . 7 ⊢ Ⅎ𝑥𝐷
8		nfcv 2901	. . . . . . . . 9 ⊢ Ⅎ𝑥𝑦
9	5, 8	nffv 6900	. . . . . . . 8 ⊢ Ⅎ𝑥(𝐹‘𝑦)
10		nfcv 2901	. . . . . . . 8 ⊢ Ⅎ𝑥 <
11		nfcv 2901	. . . . . . . 8 ⊢ Ⅎ𝑥𝑎
12	9, 10, 11	nfbr 5194	. . . . . . 7 ⊢ Ⅎ𝑥(𝐹‘𝑦) < 𝑎
13		nfv 1915	. . . . . . 7 ⊢ Ⅎ𝑦(𝐹‘𝑥) < 𝑎
14		fveq2 6890	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝐹‘𝑦) = (𝐹‘𝑥))
15	14	breq1d 5157	. . . . . . 7 ⊢ (𝑦 = 𝑥 → ((𝐹‘𝑦) < 𝑎 ↔ (𝐹‘𝑥) < 𝑎))
16	4, 7, 12, 13, 15	cbvrabw 3465	. . . . . 6 ⊢ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) < 𝑎} = {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎}
17	16	eleq1i 2822	. . . . 5 ⊢ ({𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) < 𝑎} ∈ (𝑆 ↾t 𝐷) ↔ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷))
18	17	ralbii 3091	. . . 4 ⊢ (∀𝑎 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) < 𝑎} ∈ (𝑆 ↾t 𝐷) ↔ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷))
19	18	3anbi3i 1157	. . 3 ⊢ ((𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) < 𝑎} ∈ (𝑆 ↾t 𝐷)) ↔ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷)))
20	19	a1i 11	. 2 ⊢ (𝜑 → ((𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) < 𝑎} ∈ (𝑆 ↾t 𝐷)) ↔ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷))))
21	3, 20	bitrd 278	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 205   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104  Ⅎwnfc 2881  ∀wral 3059  {crab 3430   ⊆ wss 3947  ∪ cuni 4907   class class class wbr 5147  dom cdm 5675  ⟶wf 6538  ‘cfv 6542  (class class class)co 7411  ℝcr 11111   < clt 11252   ↾_t crest 17370  SAlgcsalg 45322  SMblFncsmblfn 45709

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-pre-lttri 11186  ax-pre-lttrn 11187

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-po 5587  df-so 5588  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7977  df-2nd 7978  df-er 8705  df-pm 8825  df-en 8942  df-dom 8943  df-sdom 8944  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-ioo 13332  df-ico 13334  df-smblfn 45710

This theorem is referenced by:  smfpreimaltf  45750  issmfdf  45751