Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > smfpimgtmpt | Structured version Visualization version GIF version |
Description: Given a function measurable w.r.t. to a sigma-algebra, the preimage of an open interval unbounded above is in the subspace sigma-algebra induced by its domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
smfpimgtmpt.x | ⊢ Ⅎ𝑥𝜑 |
smfpimgtmpt.s | ⊢ (𝜑 → 𝑆 ∈ SAlg) |
smfpimgtmpt.b | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) |
smfpimgtmpt.f | ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆)) |
smfpimgtmpt.l | ⊢ (𝜑 → 𝐿 ∈ ℝ) |
Ref | Expression |
---|---|
smfpimgtmpt | ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐿 < 𝐵} ∈ (𝑆 ↾t 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfmpt1 5166 | . . 3 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵) | |
2 | smfpimgtmpt.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ SAlg) | |
3 | smfpimgtmpt.f | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆)) | |
4 | eqid 2823 | . . 3 ⊢ dom (𝑥 ∈ 𝐴 ↦ 𝐵) = dom (𝑥 ∈ 𝐴 ↦ 𝐵) | |
5 | smfpimgtmpt.l | . . 3 ⊢ (𝜑 → 𝐿 ∈ ℝ) | |
6 | 1, 2, 3, 4, 5 | smfpreimagtf 43051 | . 2 ⊢ (𝜑 → {𝑥 ∈ dom (𝑥 ∈ 𝐴 ↦ 𝐵) ∣ 𝐿 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)} ∈ (𝑆 ↾t dom (𝑥 ∈ 𝐴 ↦ 𝐵))) |
7 | smfpimgtmpt.x | . . . . . 6 ⊢ Ⅎ𝑥𝜑 | |
8 | eqid 2823 | . . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
9 | smfpimgtmpt.b | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) | |
10 | 7, 8, 9 | dmmptdf 41495 | . . . . 5 ⊢ (𝜑 → dom (𝑥 ∈ 𝐴 ↦ 𝐵) = 𝐴) |
11 | 1 | nfdm 5825 | . . . . . 6 ⊢ Ⅎ𝑥dom (𝑥 ∈ 𝐴 ↦ 𝐵) |
12 | nfcv 2979 | . . . . . 6 ⊢ Ⅎ𝑥𝐴 | |
13 | 11, 12 | rabeqf 3483 | . . . . 5 ⊢ (dom (𝑥 ∈ 𝐴 ↦ 𝐵) = 𝐴 → {𝑥 ∈ dom (𝑥 ∈ 𝐴 ↦ 𝐵) ∣ 𝐿 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)} = {𝑥 ∈ 𝐴 ∣ 𝐿 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)}) |
14 | 10, 13 | syl 17 | . . . 4 ⊢ (𝜑 → {𝑥 ∈ dom (𝑥 ∈ 𝐴 ↦ 𝐵) ∣ 𝐿 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)} = {𝑥 ∈ 𝐴 ∣ 𝐿 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)}) |
15 | 8 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)) |
16 | 15, 9 | fvmpt2d 6783 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵) |
17 | 16 | breq2d 5080 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐿 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ↔ 𝐿 < 𝐵)) |
18 | 7, 17 | rabbida 3476 | . . . 4 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐿 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)} = {𝑥 ∈ 𝐴 ∣ 𝐿 < 𝐵}) |
19 | eqidd 2824 | . . . 4 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐿 < 𝐵} = {𝑥 ∈ 𝐴 ∣ 𝐿 < 𝐵}) | |
20 | 14, 18, 19 | 3eqtrrd 2863 | . . 3 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐿 < 𝐵} = {𝑥 ∈ dom (𝑥 ∈ 𝐴 ↦ 𝐵) ∣ 𝐿 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)}) |
21 | 10 | eqcomd 2829 | . . . 4 ⊢ (𝜑 → 𝐴 = dom (𝑥 ∈ 𝐴 ↦ 𝐵)) |
22 | 21 | oveq2d 7174 | . . 3 ⊢ (𝜑 → (𝑆 ↾t 𝐴) = (𝑆 ↾t dom (𝑥 ∈ 𝐴 ↦ 𝐵))) |
23 | 20, 22 | eleq12d 2909 | . 2 ⊢ (𝜑 → ({𝑥 ∈ 𝐴 ∣ 𝐿 < 𝐵} ∈ (𝑆 ↾t 𝐴) ↔ {𝑥 ∈ dom (𝑥 ∈ 𝐴 ↦ 𝐵) ∣ 𝐿 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)} ∈ (𝑆 ↾t dom (𝑥 ∈ 𝐴 ↦ 𝐵)))) |
24 | 6, 23 | mpbird 259 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐿 < 𝐵} ∈ (𝑆 ↾t 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 Ⅎwnf 1784 ∈ wcel 2114 {crab 3144 class class class wbr 5068 ↦ cmpt 5148 dom cdm 5557 ‘cfv 6357 (class class class)co 7158 ℝcr 10538 < clt 10677 ↾t crest 16696 SAlgcsalg 42600 SMblFncsmblfn 42984 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-cc 9859 ax-ac2 9887 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-map 8410 df-pm 8411 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-sup 8908 df-inf 8909 df-card 9370 df-acn 9373 df-ac 9544 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-n0 11901 df-z 11985 df-uz 12247 df-q 12352 df-rp 12393 df-ioo 12745 df-ico 12747 df-fl 13165 df-rest 16698 df-salg 42601 df-smblfn 42985 |
This theorem is referenced by: smfrec 43071 |
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