smfdmmblpimne - Mathbox for Glauco Siliprandi

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Theorem smfdmmblpimne 44613

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 11098	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ^*)
5		smfdmmblpimne.7	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ ℝ)
6	5	rexrd 11098	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℝ^*)
7	6	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℝ^*)
8	2, 4, 7	pimxrneun 43265	. . 3 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 ≠ 𝐶} = ({𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶} ∪ {𝑥 ∈ 𝐴 ∣ 𝐶 < 𝐵}))
9	1, 8	eqtrid 2789	. 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 44137	. . . 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 44534	. . . 4 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶} ∈ (𝑆 ↾_t 𝐴))
16	12, 15	sseldd 3932	. . 3 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶} ∈ 𝑆)
17	2, 13, 10, 3, 14, 6	smfpimgtxrmptf 44560	. . . 4 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐶 < 𝐵} ∈ (𝑆 ↾_t 𝐴))
18	12, 17	sseldd 3932	. . 3 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐶 < 𝐵} ∈ 𝑆)
19	10, 16, 18	saluncld 44124	. 2 ⊢ (𝜑 → ({𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶} ∪ {𝑥 ∈ 𝐴 ∣ 𝐶 < 𝐵}) ∈ 𝑆)
20	9, 19	eqeltrd 2838	1 ⊢ (𝜑 → 𝐷 ∈ 𝑆)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1540 Ⅎwnf 1784 ∈ wcel 2105 Ⅎwnfc 2885 ≠ wne 2941 {crab 3404 ∪ cun 3895 class class class wbr 5087 ↦ cmpt 5170 ‘cfv 6465 (class class class)co 7315 ℝcr 10943 ℝ^*cxr 11081 < clt 11082 ↾_t crest 17201 SAlgcsalg 44086 SMblFncsmblfn 44471

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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2708 ax-rep 5224 ax-sep 5238 ax-nul 5245 ax-pow 5303 ax-pr 5367 ax-un 7628 ax-inf2 9470 ax-cc 10264 ax-ac2 10292 ax-cnex 11000 ax-resscn 11001 ax-1cn 11002 ax-icn 11003 ax-addcl 11004 ax-addrcl 11005 ax-mulcl 11006 ax-mulrcl 11007 ax-mulcom 11008 ax-addass 11009 ax-mulass 11010 ax-distr 11011 ax-i2m1 11012 ax-1ne0 11013 ax-1rid 11014 ax-rnegex 11015 ax-rrecex 11016 ax-cnre 11017 ax-pre-lttri 11018 ax-pre-lttrn 11019 ax-pre-ltadd 11020 ax-pre-mulgt0 11021 ax-pre-sup 11022

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3350 df-reu 3351 df-rab 3405 df-v 3443 df-sbc 3727 df-csb 3843 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3916 df-nul 4268 df-if 4472 df-pw 4547 df-sn 4572 df-pr 4574 df-op 4578 df-uni 4851 df-int 4893 df-iun 4939 df-iin 4940 df-br 5088 df-opab 5150 df-mpt 5171 df-tr 5205 df-id 5507 df-eprel 5513 df-po 5521 df-so 5522 df-fr 5562 df-se 5563 df-we 5564 df-xp 5613 df-rel 5614 df-cnv 5615 df-co 5616 df-dm 5617 df-rn 5618 df-res 5619 df-ima 5620 df-pred 6224 df-ord 6291 df-on 6292 df-lim 6293 df-suc 6294 df-iota 6417 df-fun 6467 df-fn 6468 df-f 6469 df-f1 6470 df-fo 6471 df-f1o 6472 df-fv 6473 df-isom 6474 df-riota 7272 df-ov 7318 df-oprab 7319 df-mpo 7320 df-om 7758 df-1st 7876 df-2nd 7877 df-frecs 8144 df-wrecs 8175 df-recs 8249 df-rdg 8288 df-1o 8344 df-er 8546 df-map 8665 df-pm 8666 df-en 8782 df-dom 8783 df-sdom 8784 df-fin 8785 df-sup 9271 df-inf 9272 df-card 9768 df-acn 9771 df-ac 9945 df-pnf 11084 df-mnf 11085 df-xr 11086 df-ltxr 11087 df-le 11088 df-sub 11280 df-neg 11281 df-div 11706 df-nn 12047 df-n0 12307 df-z 12393 df-uz 12656 df-q 12762 df-rp 12804 df-ioo 13156 df-ico 13158 df-fl 13585 df-rest 17203 df-salg 44087 df-smblfn 44472

This theorem is referenced by: smfdivdmmbl 44614

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