smfpimioompt - Mathbox for Glauco Siliprandi

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Theorem smfpimioompt 47214

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 2736	. . . . . . 7 ⊢ dom (𝑥 ∈ 𝐴 ↦ 𝐵) = dom (𝑥 ∈ 𝐴 ↦ 𝐵)
7	4, 5, 6	smff 47160	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵):dom (𝑥 ∈ 𝐴 ↦ 𝐵)⟶ℝ)
8		eqid 2736	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
9		smfpimioompt.b	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑊)
10	1, 8, 9	dmmptdf 45653	. . . . . . 7 ⊢ (𝜑 → dom (𝑥 ∈ 𝐴 ↦ 𝐵) = 𝐴)
11	10	feq2d 6652	. . . . . 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 7065	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)
14	13	rexrd 11195	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ^*)
15	1, 2, 3, 14	pimiooltgt 47138	. 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (𝐿(,)𝑅)} = ({𝑥 ∈ 𝐴 ∣ 𝐵 < 𝑅} ∩ {𝑥 ∈ 𝐴 ∣ 𝐿 < 𝐵}))
16		smfpimioompt.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
17		eqid 2736	. . . 4 ⊢ (𝑆 ↾_t 𝐴) = (𝑆 ↾_t 𝐴)
18	4, 16, 17	subsalsal 46787	. . 3 ⊢ (𝜑 → (𝑆 ↾_t 𝐴) ∈ SAlg)
19	1, 4, 9, 5, 3	smfpimltxrmpt 47187	. . 3 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝑅} ∈ (𝑆 ↾_t 𝐴))
20	1, 4, 9, 5, 2	smfpimgtxrmpt 47213	. . 3 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐿 < 𝐵} ∈ (𝑆 ↾_t 𝐴))
21	18, 19, 20	salincld 46780	. 2 ⊢ (𝜑 → ({𝑥 ∈ 𝐴 ∣ 𝐵 < 𝑅} ∩ {𝑥 ∈ 𝐴 ∣ 𝐿 < 𝐵}) ∈ (𝑆 ↾_t 𝐴))
22	15, 21	eqeltrd 2836	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (𝐿(,)𝑅)} ∈ (𝑆 ↾_t 𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 Ⅎwnf 1785 ∈ wcel 2114 {crab 3389 ∩ cin 3888 class class class wbr 5085 ↦ cmpt 5166 dom cdm 5631 ⟶wf 6494 ‘cfv 6498 (class class class)co 7367 ℝcr 11037 ℝ^*cxr 11178 < clt 11179 (,)cioo 13298 ↾_t crest 17383 SAlgcsalg 46736 SMblFncsmblfn 47123

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-inf2 9562 ax-cc 10357 ax-ac2 10385 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-iin 4936 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-isom 6507 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-om 7818 df-1st 7942 df-2nd 7943 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-2o 8406 df-er 8643 df-map 8775 df-pm 8776 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-sup 9355 df-inf 9356 df-card 9863 df-acn 9866 df-ac 10038 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-n0 12438 df-z 12525 df-uz 12789 df-q 12899 df-rp 12943 df-ioo 13302 df-ico 13304 df-fl 13751 df-rest 17385 df-salg 46737 df-smblfn 47124

This theorem is referenced by: smfpimioo 47215 smfresal 47216 smfrec 47217 smfmullem4 47222

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