smfdmmblpimne - Mathbox for Glauco Siliprandi

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Theorem smfdmmblpimne 46828

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 11165	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ^*)
5		smfdmmblpimne.7	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ ℝ)
6	5	rexrd 11165	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℝ^*)
7	6	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℝ^*)
8	2, 4, 7	pimxrneun 45477	. . 3 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 ≠ 𝐶} = ({𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶} ∪ {𝑥 ∈ 𝐴 ∣ 𝐶 < 𝐵}))
9	1, 8	eqtrid 2776	. 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 46352	. . . 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 46749	. . . 4 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶} ∈ (𝑆 ↾_t 𝐴))
16	12, 15	sseldd 3936	. . 3 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶} ∈ 𝑆)
17	2, 13, 10, 3, 14, 6	smfpimgtxrmptf 46775	. . . 4 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐶 < 𝐵} ∈ (𝑆 ↾_t 𝐴))
18	12, 17	sseldd 3936	. . 3 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐶 < 𝐵} ∈ 𝑆)
19	10, 16, 18	saluncld 46339	. 2 ⊢ (𝜑 → ({𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶} ∪ {𝑥 ∈ 𝐴 ∣ 𝐶 < 𝐵}) ∈ 𝑆)
20	9, 19	eqeltrd 2828	1 ⊢ (𝜑 → 𝐷 ∈ 𝑆)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 Ⅎwnf 1783 ∈ wcel 2109 Ⅎwnfc 2876 ≠ wne 2925 {crab 3394 ∪ cun 3901 class class class wbr 5092 ↦ cmpt 5173 ‘cfv 6482 (class class class)co 7349 ℝcr 11008 ℝ^*cxr 11148 < clt 11149 ↾_t crest 17324 SAlgcsalg 46299 SMblFncsmblfn 46686

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 5218 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-inf2 9537 ax-cc 10329 ax-ac2 10357 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-pre-sup 11087

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 3343 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4859 df-int 4897 df-iun 4943 df-iin 4944 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 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 6249 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-isom 6491 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-om 7800 df-1st 7924 df-2nd 7925 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-1o 8388 df-2o 8389 df-er 8625 df-map 8755 df-pm 8756 df-en 8873 df-dom 8874 df-sdom 8875 df-fin 8876 df-sup 9332 df-inf 9333 df-card 9835 df-acn 9838 df-ac 10010 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-sub 11349 df-neg 11350 df-div 11778 df-nn 12129 df-n0 12385 df-z 12472 df-uz 12736 df-q 12850 df-rp 12894 df-ioo 13252 df-ico 13254 df-fl 13696 df-rest 17326 df-salg 46300 df-smblfn 46687

This theorem is referenced by: smfdivdmmbl 46829

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