smfdmmblpimne - Mathbox for Glauco Siliprandi

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Theorem smfdmmblpimne 46758

Description: If a measurable function w.r.t. to a sigma-algebra has domain in the sigma-algebra, the set of elements that are not mapped to a given real, is in the sigma-algebra (Contributed by Glauco Siliprandi, 5-Jan-2025.)

**Hypotheses**
Ref	Expression
smfdmmblpimne.1	⊢ Ⅎ𝑥𝜑
smfdmmblpimne.2	⊢ Ⅎ𝑥𝐴
smfdmmblpimne.3	⊢ (𝜑 → 𝑆 ∈ SAlg)
smfdmmblpimne.4	⊢ (𝜑 → 𝐴 ∈ 𝑆)
smfdmmblpimne.5	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)
smfdmmblpimne.6	⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆))
smfdmmblpimne.7	⊢ (𝜑 → 𝐶 ∈ ℝ)
smfdmmblpimne.8	⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ 𝐵 ≠ 𝐶}

**Assertion**
Ref	Expression
smfdmmblpimne	⊢ (𝜑 → 𝐷 ∈ 𝑆)

Distinct variable group: 𝑥,𝐶 Allowed substitution hints: 𝜑(𝑥) 𝐴(𝑥) 𝐵(𝑥) 𝐷(𝑥) 𝑆(𝑥)

**Proof of Theorem smfdmmblpimne**
Step	Hyp	Ref	Expression
1		smfdmmblpimne.8	. . 3 ⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ 𝐵 ≠ 𝐶}
2		smfdmmblpimne.1	. . . 4 ⊢ Ⅎ𝑥𝜑
3		smfdmmblpimne.5	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)
4	3	rexrd 11340	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ^*)
5		smfdmmblpimne.7	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ ℝ)
6	5	rexrd 11340	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℝ^*)
7	6	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℝ^*)
8	2, 4, 7	pimxrneun 45404	. . 3 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 ≠ 𝐶} = ({𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶} ∪ {𝑥 ∈ 𝐴 ∣ 𝐶 < 𝐵}))
9	1, 8	eqtrid 2792	. 2 ⊢ (𝜑 → 𝐷 = ({𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶} ∪ {𝑥 ∈ 𝐴 ∣ 𝐶 < 𝐵}))
10		smfdmmblpimne.3	. . 3 ⊢ (𝜑 → 𝑆 ∈ SAlg)
11		smfdmmblpimne.4	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑆)
12	10, 11	salrestss 46282	. . . 4 ⊢ (𝜑 → (𝑆 ↾_t 𝐴) ⊆ 𝑆)
13		smfdmmblpimne.2	. . . . 5 ⊢ Ⅎ𝑥𝐴
14		smfdmmblpimne.6	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆))
15	2, 13, 10, 3, 14, 6	smfpimltxrmptf 46679	. . . 4 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶} ∈ (𝑆 ↾_t 𝐴))
16	12, 15	sseldd 4009	. . 3 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶} ∈ 𝑆)
17	2, 13, 10, 3, 14, 6	smfpimgtxrmptf 46705	. . . 4 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐶 < 𝐵} ∈ (𝑆 ↾_t 𝐴))
18	12, 17	sseldd 4009	. . 3 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐶 < 𝐵} ∈ 𝑆)
19	10, 16, 18	saluncld 46269	. 2 ⊢ (𝜑 → ({𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶} ∪ {𝑥 ∈ 𝐴 ∣ 𝐶 < 𝐵}) ∈ 𝑆)
20	9, 19	eqeltrd 2844	1 ⊢ (𝜑 → 𝐷 ∈ 𝑆)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 Ⅎwnf 1781 ∈ wcel 2108 Ⅎwnfc 2893 ≠ wne 2946 {crab 3443 ∪ cun 3974 class class class wbr 5166 ↦ cmpt 5249 ‘cfv 6573 (class class class)co 7448 ℝcr 11183 ℝ^*cxr 11323 < clt 11324 ↾_t crest 17480 SAlgcsalg 46229 SMblFncsmblfn 46616

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-inf2 9710 ax-cc 10504 ax-ac2 10532 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-1st 8030 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-er 8763 df-map 8886 df-pm 8887 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-sup 9511 df-inf 9512 df-card 10008 df-acn 10011 df-ac 10185 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-n0 12554 df-z 12640 df-uz 12904 df-q 13014 df-rp 13058 df-ioo 13411 df-ico 13413 df-fl 13843 df-rest 17482 df-salg 46230 df-smblfn 46617

This theorem is referenced by: smfdivdmmbl 46759

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