smfdmmblpimne - Mathbox for Glauco Siliprandi

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Theorem smfdmmblpimne 47199

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 11194	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ^*)
5		smfdmmblpimne.7	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ ℝ)
6	5	rexrd 11194	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℝ^*)
7	6	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℝ^*)
8	2, 4, 7	pimxrneun 45850	. . 3 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 ≠ 𝐶} = ({𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶} ∪ {𝑥 ∈ 𝐴 ∣ 𝐶 < 𝐵}))
9	1, 8	eqtrid 2784	. 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 46723	. . . 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 47120	. . . 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 47146	. . . 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 46710	. 2 ⊢ (𝜑 → ({𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶} ∪ {𝑥 ∈ 𝐴 ∣ 𝐶 < 𝐵}) ∈ 𝑆)
20	9, 19	eqeltrd 2837	1 ⊢ (𝜑 → 𝐷 ∈ 𝑆)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1542 Ⅎwnf 1785 ∈ wcel 2114 Ⅎwnfc 2884 ≠ wne 2933 {crab 3401 ∪ cun 3901 class class class wbr 5100 ↦ cmpt 5181 ‘cfv 6500 (class class class)co 7368 ℝcr 11037 ℝ^*cxr 11177 < clt 11178 ↾_t crest 17352 SAlgcsalg 46670 SMblFncsmblfn 47057

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-inf2 9562 ax-cc 10357 ax-ac2 10385 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-iin 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-se 5586 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-om 7819 df-1st 7943 df-2nd 7944 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-2o 8408 df-er 8645 df-map 8777 df-pm 8778 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-sup 9357 df-inf 9358 df-card 9863 df-acn 9866 df-ac 10038 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-div 11807 df-nn 12158 df-n0 12414 df-z 12501 df-uz 12764 df-q 12874 df-rp 12918 df-ioo 13277 df-ico 13279 df-fl 13724 df-rest 17354 df-salg 46671 df-smblfn 47058

This theorem is referenced by: smfdivdmmbl 47200

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