smfdmmblpimne - Mathbox for Glauco Siliprandi

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Theorem smfdmmblpimne 46494

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 11305	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ^*)
5		smfdmmblpimne.7	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ ℝ)
6	5	rexrd 11305	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℝ^*)
7	6	adantr 479	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℝ^*)
8	2, 4, 7	pimxrneun 45140	. . 3 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 ≠ 𝐶} = ({𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶} ∪ {𝑥 ∈ 𝐴 ∣ 𝐶 < 𝐵}))
9	1, 8	eqtrid 2778	. 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 46018	. . . 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 46415	. . . 4 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶} ∈ (𝑆 ↾_t 𝐴))
16	12, 15	sseldd 3979	. . 3 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶} ∈ 𝑆)
17	2, 13, 10, 3, 14, 6	smfpimgtxrmptf 46441	. . . 4 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐶 < 𝐵} ∈ (𝑆 ↾_t 𝐴))
18	12, 17	sseldd 3979	. . 3 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐶 < 𝐵} ∈ 𝑆)
19	10, 16, 18	saluncld 46005	. 2 ⊢ (𝜑 → ({𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶} ∪ {𝑥 ∈ 𝐴 ∣ 𝐶 < 𝐵}) ∈ 𝑆)
20	9, 19	eqeltrd 2826	1 ⊢ (𝜑 → 𝐷 ∈ 𝑆)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 Ⅎwnf 1778 ∈ wcel 2099 Ⅎwnfc 2876 ≠ wne 2930 {crab 3419 ∪ cun 3944 class class class wbr 5145 ↦ cmpt 5228 ‘cfv 6546 (class class class)co 7416 ℝcr 11148 ℝ^*cxr 11288 < clt 11289 ↾_t crest 17430 SAlgcsalg 45965 SMblFncsmblfn 46352

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5282 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 ax-inf2 9677 ax-cc 10469 ax-ac2 10497 ax-cnex 11205 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 ax-pre-mulgt0 11226 ax-pre-sup 11227

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-int 4947 df-iun 4995 df-iin 4996 df-br 5146 df-opab 5208 df-mpt 5229 df-tr 5263 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-se 5630 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6304 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-isom 6555 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7995 df-2nd 7996 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-1o 8488 df-2o 8489 df-er 8726 df-map 8849 df-pm 8850 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-sup 9478 df-inf 9479 df-card 9975 df-acn 9978 df-ac 10152 df-pnf 11291 df-mnf 11292 df-xr 11293 df-ltxr 11294 df-le 11295 df-sub 11487 df-neg 11488 df-div 11913 df-nn 12259 df-n0 12519 df-z 12605 df-uz 12869 df-q 12979 df-rp 13023 df-ioo 13376 df-ico 13378 df-fl 13806 df-rest 17432 df-salg 45966 df-smblfn 46353

This theorem is referenced by: smfdivdmmbl 46495

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