Theorem issmff 47713

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 47707	. 2 ⊢ (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) < 𝑎} ∈ (𝑆 ↾t 𝐷))))
4		nfcv 2923	. . . . . . 7 ⊢ Ⅎ𝑦𝐷
5		issmff.x	. . . . . . . . 9 ⊢ Ⅎ𝑥𝐹
6	5	nfdm 5933	. . . . . . . 8 ⊢ Ⅎ𝑥dom 𝐹
7	2, 6	nfcxfr 2921	. . . . . . 7 ⊢ Ⅎ𝑥𝐷
8		nfcv 2923	. . . . . . . . 9 ⊢ Ⅎ𝑥𝑦
9	5, 8	nffv 6893	. . . . . . . 8 ⊢ Ⅎ𝑥(𝐹‘𝑦)
10		nfcv 2923	. . . . . . . 8 ⊢ Ⅎ𝑥 <
11		nfcv 2923	. . . . . . . 8 ⊢ Ⅎ𝑥𝑎
12	9, 10, 11	nfbr 5152	. . . . . . 7 ⊢ Ⅎ𝑥(𝐹‘𝑦) < 𝑎
13		nfv 1947	. . . . . . 7 ⊢ Ⅎ𝑦(𝐹‘𝑥) < 𝑎
14		fveq2 6883	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝐹‘𝑦) = (𝐹‘𝑥))
15	14	breq1d 5113	. . . . . . 7 ⊢ (𝑦 = 𝑥 → ((𝐹‘𝑦) < 𝑎 ↔ (𝐹‘𝑥) < 𝑎))
16	4, 7, 12, 13, 15	cbvrabw 3447	. . . . . 6 ⊢ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) < 𝑎} = {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎}
17	16	eleq1i 2852	. . . . 5 ⊢ ({𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) < 𝑎} ∈ (𝑆 ↾t 𝐷) ↔ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷))
18	17	ralbii 3109	. . . 4 ⊢ (∀𝑎 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) < 𝑎} ∈ (𝑆 ↾t 𝐷) ↔ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷))
19	18	3anbi3i 1177	. . 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 282	1 ⊢ (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  Ⅎwnfc 2908  ∀wral 3077  {crab 3413   ⊆ wss 3899  ∪ cuni 4867   class class class wbr 5103  dom cdm 5651  ⟶wf 6533  ‘cfv 6537  (class class class)co 7418  ℝcr 11192   < clt 11336   ↾_t crest 17584  SAlgcsalg 47287  SMblFncsmblfn 47674

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749  ax-cnex 11249  ax-resscn 11250  ax-pre-lttri 11267  ax-pre-lttrn 11268

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-po 5559  df-so 5560  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7421  df-oprab 7422  df-mpo 7423  df-1st 7999  df-2nd 8000  df-er 8710  df-pm 8843  df-en 8967  df-dom 8968  df-sdom 8969  df-pnf 11338  df-mnf 11339  df-xr 11340  df-ltxr 11341  df-le 11342  df-ioo 13473  df-ico 13475  df-smblfn 47675

This theorem is used by:  smfpreimaltf  47715  issmfdf  47716