smfpimioompt - Mathbox for Glauco Siliprandi

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Theorem smfpimioompt 46908

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 2733	. . . . . . 7 ⊢ dom (𝑥 ∈ 𝐴 ↦ 𝐵) = dom (𝑥 ∈ 𝐴 ↦ 𝐵)
7	4, 5, 6	smff 46854	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵):dom (𝑥 ∈ 𝐴 ↦ 𝐵)⟶ℝ)
8		eqid 2733	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
9		smfpimioompt.b	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑊)
10	1, 8, 9	dmmptdf 45345	. . . . . . 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 7052	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)
14	13	rexrd 11169	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ^*)
15	1, 2, 3, 14	pimiooltgt 46832	. 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (𝐿(,)𝑅)} = ({𝑥 ∈ 𝐴 ∣ 𝐵 < 𝑅} ∩ {𝑥 ∈ 𝐴 ∣ 𝐿 < 𝐵}))
16		smfpimioompt.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
17		eqid 2733	. . . 4 ⊢ (𝑆 ↾_t 𝐴) = (𝑆 ↾_t 𝐴)
18	4, 16, 17	subsalsal 46481	. . 3 ⊢ (𝜑 → (𝑆 ↾_t 𝐴) ∈ SAlg)
19	1, 4, 9, 5, 3	smfpimltxrmpt 46881	. . 3 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝑅} ∈ (𝑆 ↾_t 𝐴))
20	1, 4, 9, 5, 2	smfpimgtxrmpt 46907	. . 3 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐿 < 𝐵} ∈ (𝑆 ↾_t 𝐴))
21	18, 19, 20	salincld 46474	. 2 ⊢ (𝜑 → ({𝑥 ∈ 𝐴 ∣ 𝐵 < 𝑅} ∩ {𝑥 ∈ 𝐴 ∣ 𝐿 < 𝐵}) ∈ (𝑆 ↾_t 𝐴))
22	15, 21	eqeltrd 2833	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (𝐿(,)𝑅)} ∈ (𝑆 ↾_t 𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 Ⅎwnf 1784 ∈ wcel 2113 {crab 3396 ∩ cin 3897 class class class wbr 5093 ↦ cmpt 5174 dom cdm 5619 ⟶wf 6482 ‘cfv 6486 (class class class)co 7352 ℝcr 11012 ℝ^*cxr 11152 < clt 11153 (,)cioo 13247 ↾_t crest 17326 SAlgcsalg 46430 SMblFncsmblfn 46817

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 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-rep 5219 ax-sep 5236 ax-nul 5246 ax-pow 5305 ax-pr 5372 ax-un 7674 ax-inf2 9538 ax-cc 10333 ax-ac2 10361 ax-cnex 11069 ax-resscn 11070 ax-1cn 11071 ax-icn 11072 ax-addcl 11073 ax-addrcl 11074 ax-mulcl 11075 ax-mulrcl 11076 ax-mulcom 11077 ax-addass 11078 ax-mulass 11079 ax-distr 11080 ax-i2m1 11081 ax-1ne0 11082 ax-1rid 11083 ax-rnegex 11084 ax-rrecex 11085 ax-cnre 11086 ax-pre-lttri 11087 ax-pre-lttrn 11088 ax-pre-ltadd 11089 ax-pre-mulgt0 11090 ax-pre-sup 11091

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-rmo 3347 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4283 df-if 4475 df-pw 4551 df-sn 4576 df-pr 4578 df-op 4582 df-uni 4859 df-int 4898 df-iun 4943 df-iin 4944 df-br 5094 df-opab 5156 df-mpt 5175 df-tr 5201 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-se 5573 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 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 7309 df-ov 7355 df-oprab 7356 df-mpo 7357 df-om 7803 df-1st 7927 df-2nd 7928 df-frecs 8217 df-wrecs 8248 df-recs 8297 df-rdg 8335 df-1o 8391 df-2o 8392 df-er 8628 df-map 8758 df-pm 8759 df-en 8876 df-dom 8877 df-sdom 8878 df-fin 8879 df-sup 9333 df-inf 9334 df-card 9839 df-acn 9842 df-ac 10014 df-pnf 11155 df-mnf 11156 df-xr 11157 df-ltxr 11158 df-le 11159 df-sub 11353 df-neg 11354 df-div 11782 df-nn 12133 df-n0 12389 df-z 12476 df-uz 12739 df-q 12849 df-rp 12893 df-ioo 13251 df-ico 13253 df-fl 13698 df-rest 17328 df-salg 46431 df-smblfn 46818

This theorem is referenced by: smfpimioo 46909 smfresal 46910 smfrec 46911 smfmullem4 46916

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