Theorem issmfgtlem 43830

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 42208	. . . . . . . 8 ⊢ (𝜑 → ∪ (𝑆 ↾t 𝐷) = 𝐷)
5	4	eqcomd 2744	. . . . . . 7 ⊢ (𝜑 → 𝐷 = ∪ (𝑆 ↾t 𝐷))
6	5	rabeqdv 3386	. . . . . 6 ⊢ (𝜑 → {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑏} = {𝑥 ∈ ∪ (𝑆 ↾t 𝐷) ∣ (𝐹‘𝑥) < 𝑏})
7	6	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ ℝ) → {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑏} = {𝑥 ∈ ∪ (𝑆 ↾t 𝐷) ∣ (𝐹‘𝑥) < 𝑏})
8		issmfgtlem.x	. . . . . . 7 ⊢ Ⅎ𝑥𝜑
9		nfv 1921	. . . . . . 7 ⊢ Ⅎ𝑥 𝑏 ∈ ℝ
10	8, 9	nfan 1906	. . . . . 6 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑏 ∈ ℝ)
11		issmfgtlem.a	. . . . . . 7 ⊢ Ⅎ𝑎𝜑
12		nfv 1921	. . . . . . 7 ⊢ Ⅎ𝑎 𝑏 ∈ ℝ
13	11, 12	nfan 1906	. . . . . 6 ⊢ Ⅎ𝑎(𝜑 ∧ 𝑏 ∈ ℝ)
14	3	uniexd 7486	. . . . . . . . . . 11 ⊢ (𝜑 → ∪ 𝑆 ∈ V)
15	14	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐷 ⊆ ∪ 𝑆) → ∪ 𝑆 ∈ V)
16		simpr 488	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐷 ⊆ ∪ 𝑆) → 𝐷 ⊆ ∪ 𝑆)
17	15, 16	ssexd 5192	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐷 ⊆ ∪ 𝑆) → 𝐷 ∈ V)
18	1, 17	mpdan 687	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ V)
19		eqid 2738	. . . . . . . 8 ⊢ (𝑆 ↾t 𝐷) = (𝑆 ↾t 𝐷)
20	3, 18, 19	subsalsal 43440	. . . . . . 7 ⊢ (𝜑 → (𝑆 ↾t 𝐷) ∈ SAlg)
21	20	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ ℝ) → (𝑆 ↾t 𝐷) ∈ SAlg)
22		eqid 2738	. . . . . 6 ⊢ ∪ (𝑆 ↾t 𝐷) = ∪ (𝑆 ↾t 𝐷)
23	2	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝑆 ↾t 𝐷)) → 𝐹:𝐷⟶ℝ)
24		simpr 488	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝑆 ↾t 𝐷)) → 𝑥 ∈ ∪ (𝑆 ↾t 𝐷))
25	4	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝑆 ↾t 𝐷)) → ∪ (𝑆 ↾t 𝐷) = 𝐷)
26	24, 25	eleqtrd 2835	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝑆 ↾t 𝐷)) → 𝑥 ∈ 𝐷)
27	23, 26	ffvelrnd 6862	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝑆 ↾t 𝐷)) → (𝐹‘𝑥) ∈ ℝ)
28	27	rexrd 10769	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝑆 ↾t 𝐷)) → (𝐹‘𝑥) ∈ ℝ*)
29	28	adantlr 715	. . . . . 6 ⊢ (((𝜑 ∧ 𝑏 ∈ ℝ) ∧ 𝑥 ∈ ∪ (𝑆 ↾t 𝐷)) → (𝐹‘𝑥) ∈ ℝ*)
30	4	rabeqdv 3386	. . . . . . . . 9 ⊢ (𝜑 → {𝑥 ∈ ∪ (𝑆 ↾t 𝐷) ∣ 𝑎 < (𝐹‘𝑥)} = {𝑥 ∈ 𝐷 ∣ 𝑎 < (𝐹‘𝑥)})
31	30	adantr 484	. . . . . . . 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 3121	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐷 ∣ 𝑎 < (𝐹‘𝑥)} ∈ (𝑆 ↾t 𝐷))
34	31, 33	eqeltrd 2833	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ ∪ (𝑆 ↾t 𝐷) ∣ 𝑎 < (𝐹‘𝑥)} ∈ (𝑆 ↾t 𝐷))
35	34	adantlr 715	. . . . . 6 ⊢ (((𝜑 ∧ 𝑏 ∈ ℝ) ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ ∪ (𝑆 ↾t 𝐷) ∣ 𝑎 < (𝐹‘𝑥)} ∈ (𝑆 ↾t 𝐷))
36		simpr 488	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ ℝ) → 𝑏 ∈ ℝ)
37	10, 13, 21, 22, 29, 35, 36	salpreimagtlt 43805	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ ℝ) → {𝑥 ∈ ∪ (𝑆 ↾t 𝐷) ∣ (𝐹‘𝑥) < 𝑏} ∈ (𝑆 ↾t 𝐷))
38	7, 37	eqeltrd 2833	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ ℝ) → {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑏} ∈ (𝑆 ↾t 𝐷))
39	38	ralrimiva 3096	. . 3 ⊢ (𝜑 → ∀𝑏 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑏} ∈ (𝑆 ↾t 𝐷))
40	1, 2, 39	3jca 1129	. 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 43803	. 2 ⊢ (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑏} ∈ (𝑆 ↾t 𝐷))))
43	40, 42	mpbird 260	1 ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆))

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1088   = wceq 1542  Ⅎwnf 1790   ∈ wcel 2114  ∀wral 3053  {crab 3057  Vcvv 3398   ⊆ wss 3843  ∪ cuni 4796   class class class wbr 5030  dom cdm 5525  ⟶wf 6335  ‘cfv 6339  (class class class)co 7170  ℝcr 10614  ℝ^*cxr 10752   < clt 10753   ↾_t crest 16797  SAlgcsalg 43391  SMblFncsmblfn 43775

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479  ax-inf2 9177  ax-cc 9935  ax-ac2 9963  ax-cnex 10671  ax-resscn 10672  ax-1cn 10673  ax-icn 10674  ax-addcl 10675  ax-addrcl 10676  ax-mulcl 10677  ax-mulrcl 10678  ax-mulcom 10679  ax-addass 10680  ax-mulass 10681  ax-distr 10682  ax-i2m1 10683  ax-1ne0 10684  ax-1rid 10685  ax-rnegex 10686  ax-rrecex 10687  ax-cnre 10688  ax-pre-lttri 10689  ax-pre-lttrn 10690  ax-pre-ltadd 10691  ax-pre-mulgt0 10692  ax-pre-sup 10693

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-nel 3039  df-ral 3058  df-rex 3059  df-reu 3060  df-rmo 3061  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-tp 4521  df-op 4523  df-uni 4797  df-int 4837  df-iun 4883  df-iin 4884  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-se 5484  df-we 5485  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6129  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-isom 6348  df-riota 7127  df-ov 7173  df-oprab 7174  df-mpo 7175  df-om 7600  df-1st 7714  df-2nd 7715  df-wrecs 7976  df-recs 8037  df-rdg 8075  df-1o 8131  df-er 8320  df-map 8439  df-pm 8440  df-en 8556  df-dom 8557  df-sdom 8558  df-fin 8559  df-sup 8979  df-inf 8980  df-card 9441  df-acn 9444  df-ac 9616  df-pnf 10755  df-mnf 10756  df-xr 10757  df-ltxr 10758  df-le 10759  df-sub 10950  df-neg 10951  df-div 11376  df-nn 11717  df-n0 11977  df-z 12063  df-uz 12325  df-q 12431  df-rp 12473  df-ioo 12825  df-ico 12827  df-fl 13253  df-rest 16799  df-salg 43392  df-smblfn 43776

This theorem is referenced by:  issmfgt  43831