Theorem issmfgtlem 43052

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-open intervals unbounded above 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 (iii) of [Fremlin1] p. 35 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypotheses

Ref	Expression
issmfgtlem.x	⊢ Ⅎ𝑥𝜑
issmfgtlem.a	⊢ Ⅎ𝑎𝜑
issmfgtlem.s	⊢ (𝜑 → 𝑆 ∈ SAlg)
issmfgtlem.d	⊢ 𝐷 = dom 𝐹
issmfgtlem.i	⊢ (𝜑 → 𝐷 ⊆ ∪ 𝑆)
issmfgtlem.f	⊢ (𝜑 → 𝐹:𝐷⟶ℝ)
issmfgtlem.p	⊢ (𝜑 → ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ 𝑎 < (𝐹‘𝑥)} ∈ (𝑆 ↾t 𝐷))

Assertion

Ref	Expression
issmfgtlem	⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆))

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

Allowed substitution hints:   𝜑(𝑥,𝑎)

Proof of Theorem issmfgtlem

Dummy variable 𝑏 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		issmfgtlem.i	. . 3 ⊢ (𝜑 → 𝐷 ⊆ ∪ 𝑆)
2		issmfgtlem.f	. . 3 ⊢ (𝜑 → 𝐹:𝐷⟶ℝ)
3		issmfgtlem.s	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ SAlg)
4	3, 1	restuni4 41407	. . . . . . . 8 ⊢ (𝜑 → ∪ (𝑆 ↾t 𝐷) = 𝐷)
5	4	eqcomd 2827	. . . . . . 7 ⊢ (𝜑 → 𝐷 = ∪ (𝑆 ↾t 𝐷))
6	5	rabeqdv 3484	. . . . . 6 ⊢ (𝜑 → {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑏} = {𝑥 ∈ ∪ (𝑆 ↾t 𝐷) ∣ (𝐹‘𝑥) < 𝑏})
7	6	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ ℝ) → {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑏} = {𝑥 ∈ ∪ (𝑆 ↾t 𝐷) ∣ (𝐹‘𝑥) < 𝑏})
8		issmfgtlem.x	. . . . . . 7 ⊢ Ⅎ𝑥𝜑
9		nfv 1915	. . . . . . 7 ⊢ Ⅎ𝑥 𝑏 ∈ ℝ
10	8, 9	nfan 1900	. . . . . 6 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑏 ∈ ℝ)
11		issmfgtlem.a	. . . . . . 7 ⊢ Ⅎ𝑎𝜑
12		nfv 1915	. . . . . . 7 ⊢ Ⅎ𝑎 𝑏 ∈ ℝ
13	11, 12	nfan 1900	. . . . . 6 ⊢ Ⅎ𝑎(𝜑 ∧ 𝑏 ∈ ℝ)
14	3	uniexd 7468	. . . . . . . . . . 11 ⊢ (𝜑 → ∪ 𝑆 ∈ V)
15	14	adantr 483	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐷 ⊆ ∪ 𝑆) → ∪ 𝑆 ∈ V)
16		simpr 487	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐷 ⊆ ∪ 𝑆) → 𝐷 ⊆ ∪ 𝑆)
17	15, 16	ssexd 5228	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐷 ⊆ ∪ 𝑆) → 𝐷 ∈ V)
18	1, 17	mpdan 685	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ V)
19		eqid 2821	. . . . . . . 8 ⊢ (𝑆 ↾t 𝐷) = (𝑆 ↾t 𝐷)
20	3, 18, 19	subsalsal 42662	. . . . . . 7 ⊢ (𝜑 → (𝑆 ↾t 𝐷) ∈ SAlg)
21	20	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ ℝ) → (𝑆 ↾t 𝐷) ∈ SAlg)
22		eqid 2821	. . . . . 6 ⊢ ∪ (𝑆 ↾t 𝐷) = ∪ (𝑆 ↾t 𝐷)
23	2	adantr 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝑆 ↾t 𝐷)) → 𝐹:𝐷⟶ℝ)
24		simpr 487	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝑆 ↾t 𝐷)) → 𝑥 ∈ ∪ (𝑆 ↾t 𝐷))
25	4	adantr 483	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝑆 ↾t 𝐷)) → ∪ (𝑆 ↾t 𝐷) = 𝐷)
26	24, 25	eleqtrd 2915	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝑆 ↾t 𝐷)) → 𝑥 ∈ 𝐷)
27	23, 26	ffvelrnd 6852	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝑆 ↾t 𝐷)) → (𝐹‘𝑥) ∈ ℝ)
28	27	rexrd 10691	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝑆 ↾t 𝐷)) → (𝐹‘𝑥) ∈ ℝ*)
29	28	adantlr 713	. . . . . 6 ⊢ (((𝜑 ∧ 𝑏 ∈ ℝ) ∧ 𝑥 ∈ ∪ (𝑆 ↾t 𝐷)) → (𝐹‘𝑥) ∈ ℝ*)
30	4	rabeqdv 3484	. . . . . . . . 9 ⊢ (𝜑 → {𝑥 ∈ ∪ (𝑆 ↾t 𝐷) ∣ 𝑎 < (𝐹‘𝑥)} = {𝑥 ∈ 𝐷 ∣ 𝑎 < (𝐹‘𝑥)})
31	30	adantr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ ∪ (𝑆 ↾t 𝐷) ∣ 𝑎 < (𝐹‘𝑥)} = {𝑥 ∈ 𝐷 ∣ 𝑎 < (𝐹‘𝑥)})
32		issmfgtlem.p	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ 𝑎 < (𝐹‘𝑥)} ∈ (𝑆 ↾t 𝐷))
33	32	r19.21bi 3208	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐷 ∣ 𝑎 < (𝐹‘𝑥)} ∈ (𝑆 ↾t 𝐷))
34	31, 33	eqeltrd 2913	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ ∪ (𝑆 ↾t 𝐷) ∣ 𝑎 < (𝐹‘𝑥)} ∈ (𝑆 ↾t 𝐷))
35	34	adantlr 713	. . . . . 6 ⊢ (((𝜑 ∧ 𝑏 ∈ ℝ) ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ ∪ (𝑆 ↾t 𝐷) ∣ 𝑎 < (𝐹‘𝑥)} ∈ (𝑆 ↾t 𝐷))
36		simpr 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ ℝ) → 𝑏 ∈ ℝ)
37	10, 13, 21, 22, 29, 35, 36	salpreimagtlt 43027	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ ℝ) → {𝑥 ∈ ∪ (𝑆 ↾t 𝐷) ∣ (𝐹‘𝑥) < 𝑏} ∈ (𝑆 ↾t 𝐷))
38	7, 37	eqeltrd 2913	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ ℝ) → {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑏} ∈ (𝑆 ↾t 𝐷))
39	38	ralrimiva 3182	. . 3 ⊢ (𝜑 → ∀𝑏 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑏} ∈ (𝑆 ↾t 𝐷))
40	1, 2, 39	3jca 1124	. 2 ⊢ (𝜑 → (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑏} ∈ (𝑆 ↾t 𝐷)))
41		issmfgtlem.d	. . 3 ⊢ 𝐷 = dom 𝐹
42	3, 41	issmf 43025	. 2 ⊢ (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑏} ∈ (𝑆 ↾t 𝐷))))
43	40, 42	mpbird 259	1 ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆))

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083   = wceq 1537  Ⅎwnf 1784   ∈ wcel 2114  ∀wral 3138  {crab 3142  Vcvv 3494   ⊆ wss 3936  ∪ cuni 4838   class class class wbr 5066  dom cdm 5555  ⟶wf 6351  ‘cfv 6355  (class class class)co 7156  ℝcr 10536  ℝ^*cxr 10674   < clt 10675   ↾_t crest 16694  SAlgcsalg 42613  SMblFncsmblfn 42997

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-inf2 9104  ax-cc 9857  ax-ac2 9885  ax-cnex 10593  ax-resscn 10594  ax-1cn 10595  ax-icn 10596  ax-addcl 10597  ax-addrcl 10598  ax-mulcl 10599  ax-mulrcl 10600  ax-mulcom 10601  ax-addass 10602  ax-mulass 10603  ax-distr 10604  ax-i2m1 10605  ax-1ne0 10606  ax-1rid 10607  ax-rnegex 10608  ax-rrecex 10609  ax-cnre 10610  ax-pre-lttri 10611  ax-pre-lttrn 10612  ax-pre-ltadd 10613  ax-pre-mulgt0 10614  ax-pre-sup 10615

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-int 4877  df-iun 4921  df-iin 4922  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-se 5515  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-isom 6364  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7581  df-1st 7689  df-2nd 7690  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-1o 8102  df-oadd 8106  df-er 8289  df-map 8408  df-pm 8409  df-en 8510  df-dom 8511  df-sdom 8512  df-fin 8513  df-sup 8906  df-inf 8907  df-card 9368  df-acn 9371  df-ac 9542  df-pnf 10677  df-mnf 10678  df-xr 10679  df-ltxr 10680  df-le 10681  df-sub 10872  df-neg 10873  df-div 11298  df-nn 11639  df-n0 11899  df-z 11983  df-uz 12245  df-q 12350  df-rp 12391  df-ioo 12743  df-ico 12745  df-fl 13163  df-rest 16696  df-salg 42614  df-smblfn 42998

This theorem is referenced by:  issmfgt  43053