smfpimioompt - Mathbox for Glauco Siliprandi

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Theorem smfpimioompt 46768

Description: Given a function measurable w.r.t. to a sigma-algebra, the preimage of an open interval is in the subspace sigma-algebra induced by its domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

**Hypotheses**
Ref	Expression
smfpimioompt.x	⊢ Ⅎ𝑥𝜑
smfpimioompt.s	⊢ (𝜑 → 𝑆 ∈ SAlg)
smfpimioompt.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
smfpimioompt.b	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑊)
smfpimioompt.m	⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆))
smfpimioompt.l	⊢ (𝜑 → 𝐿 ∈ ℝ^*)
smfpimioompt.r	⊢ (𝜑 → 𝑅 ∈ ℝ^*)

**Assertion**
Ref	Expression
smfpimioompt	⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (𝐿(,)𝑅)} ∈ (𝑆 ↾_t 𝐴))

Distinct variable groups: 𝑥,𝐴 𝑥,𝐿 𝑥,𝑅 Allowed substitution hints: 𝜑(𝑥) 𝐵(𝑥) 𝑆(𝑥) 𝑉(𝑥) 𝑊(𝑥)

**Proof of Theorem smfpimioompt**
Step	Hyp	Ref	Expression
1		smfpimioompt.x	. . 3 ⊢ Ⅎ𝑥𝜑
2		smfpimioompt.l	. . 3 ⊢ (𝜑 → 𝐿 ∈ ℝ^*)
3		smfpimioompt.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ ℝ^*)
4		smfpimioompt.s	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ SAlg)
5		smfpimioompt.m	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆))
6		eqid 2729	. . . . . . 7 ⊢ dom (𝑥 ∈ 𝐴 ↦ 𝐵) = dom (𝑥 ∈ 𝐴 ↦ 𝐵)
7	4, 5, 6	smff 46714	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵):dom (𝑥 ∈ 𝐴 ↦ 𝐵)⟶ℝ)
8		eqid 2729	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
9		smfpimioompt.b	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑊)
10	1, 8, 9	dmmptdf 45202	. . . . . . 7 ⊢ (𝜑 → dom (𝑥 ∈ 𝐴 ↦ 𝐵) = 𝐴)
11	10	feq2d 6640	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵):dom (𝑥 ∈ 𝐴 ↦ 𝐵)⟶ℝ ↔ (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℝ))
12	7, 11	mpbid 232	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℝ)
13	12	fvmptelcdm 7051	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)
14	13	rexrd 11184	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ^*)
15	1, 2, 3, 14	pimiooltgt 46692	. 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (𝐿(,)𝑅)} = ({𝑥 ∈ 𝐴 ∣ 𝐵 < 𝑅} ∩ {𝑥 ∈ 𝐴 ∣ 𝐿 < 𝐵}))
16		smfpimioompt.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
17		eqid 2729	. . . 4 ⊢ (𝑆 ↾_t 𝐴) = (𝑆 ↾_t 𝐴)
18	4, 16, 17	subsalsal 46341	. . 3 ⊢ (𝜑 → (𝑆 ↾_t 𝐴) ∈ SAlg)
19	1, 4, 9, 5, 3	smfpimltxrmpt 46741	. . 3 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝑅} ∈ (𝑆 ↾_t 𝐴))
20	1, 4, 9, 5, 2	smfpimgtxrmpt 46767	. . 3 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐿 < 𝐵} ∈ (𝑆 ↾_t 𝐴))
21	18, 19, 20	salincld 46334	. 2 ⊢ (𝜑 → ({𝑥 ∈ 𝐴 ∣ 𝐵 < 𝑅} ∩ {𝑥 ∈ 𝐴 ∣ 𝐿 < 𝐵}) ∈ (𝑆 ↾_t 𝐴))
22	15, 21	eqeltrd 2828	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (𝐿(,)𝑅)} ∈ (𝑆 ↾_t 𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 Ⅎwnf 1783 ∈ wcel 2109 {crab 3396 ∩ cin 3904 class class class wbr 5095 ↦ cmpt 5176 dom cdm 5623 ⟶wf 6482 ‘cfv 6486 (class class class)co 7353 ℝcr 11027 ℝ^*cxr 11167 < clt 11168 (,)cioo 13266 ↾_t crest 17342 SAlgcsalg 46290 SMblFncsmblfn 46677

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 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675 ax-inf2 9556 ax-cc 10348 ax-ac2 10376 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105 ax-pre-sup 11106

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3345 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4862 df-int 4900 df-iun 4946 df-iin 4947 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-se 5577 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-isom 6495 df-riota 7310 df-ov 7356 df-oprab 7357 df-mpo 7358 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-er 8632 df-map 8762 df-pm 8763 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-sup 9351 df-inf 9352 df-card 9854 df-acn 9857 df-ac 10029 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11367 df-neg 11368 df-div 11796 df-nn 12147 df-n0 12403 df-z 12490 df-uz 12754 df-q 12868 df-rp 12912 df-ioo 13270 df-ico 13272 df-fl 13714 df-rest 17344 df-salg 46291 df-smblfn 46678

This theorem is referenced by: smfpimioo 46769 smfresal 46770 smfrec 46771 smfmullem4 46776

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