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| Mirrors > Home > MPE Home > Th. List > Mathboxes > smfpreimagtf | 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 |
|---|---|
| smfpreimagtf.x | ⊢ Ⅎ𝑥𝐹 |
| smfpreimagtf.s | ⊢ (𝜑 → 𝑆 ∈ SAlg) |
| smfpreimagtf.f | ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) |
| smfpreimagtf.d | ⊢ 𝐷 = dom 𝐹 |
| smfpreimagtf.a | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
| Ref | Expression |
|---|---|
| smfpreimagtf | ⊢ (𝜑 → {𝑥 ∈ 𝐷 ∣ 𝐴 < (𝐹‘𝑥)} ∈ (𝑆 ↾t 𝐷)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | smfpreimagtf.d | . . . . 5 ⊢ 𝐷 = dom 𝐹 | |
| 2 | smfpreimagtf.x | . . . . . 6 ⊢ Ⅎ𝑥𝐹 | |
| 3 | 2 | nfdm 5962 | . . . . 5 ⊢ Ⅎ𝑥dom 𝐹 |
| 4 | 1, 3 | nfcxfr 2903 | . . . 4 ⊢ Ⅎ𝑥𝐷 |
| 5 | nfcv 2905 | . . . 4 ⊢ Ⅎ𝑦𝐷 | |
| 6 | nfv 1914 | . . . 4 ⊢ Ⅎ𝑦 𝐴 < (𝐹‘𝑥) | |
| 7 | nfcv 2905 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
| 8 | nfcv 2905 | . . . . 5 ⊢ Ⅎ𝑥 < | |
| 9 | nfcv 2905 | . . . . . 6 ⊢ Ⅎ𝑥𝑦 | |
| 10 | 2, 9 | nffv 6916 | . . . . 5 ⊢ Ⅎ𝑥(𝐹‘𝑦) |
| 11 | 7, 8, 10 | nfbr 5190 | . . . 4 ⊢ Ⅎ𝑥 𝐴 < (𝐹‘𝑦) |
| 12 | fveq2 6906 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦)) | |
| 13 | 12 | breq2d 5155 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝐴 < (𝐹‘𝑥) ↔ 𝐴 < (𝐹‘𝑦))) |
| 14 | 4, 5, 6, 11, 13 | cbvrabw 3473 | . . 3 ⊢ {𝑥 ∈ 𝐷 ∣ 𝐴 < (𝐹‘𝑥)} = {𝑦 ∈ 𝐷 ∣ 𝐴 < (𝐹‘𝑦)} |
| 15 | 14 | a1i 11 | . 2 ⊢ (𝜑 → {𝑥 ∈ 𝐷 ∣ 𝐴 < (𝐹‘𝑥)} = {𝑦 ∈ 𝐷 ∣ 𝐴 < (𝐹‘𝑦)}) |
| 16 | smfpreimagtf.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ SAlg) | |
| 17 | smfpreimagtf.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) | |
| 18 | smfpreimagtf.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
| 19 | 16, 17, 1, 18 | smfpreimagt 46777 | . 2 ⊢ (𝜑 → {𝑦 ∈ 𝐷 ∣ 𝐴 < (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷)) |
| 20 | 15, 19 | eqeltrd 2841 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝐷 ∣ 𝐴 < (𝐹‘𝑥)} ∈ (𝑆 ↾t 𝐷)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 Ⅎwnfc 2890 {crab 3436 class class class wbr 5143 dom cdm 5685 ‘cfv 6561 (class class class)co 7431 ℝcr 11154 < clt 11295 ↾t crest 17465 SAlgcsalg 46323 SMblFncsmblfn 46710 |
| 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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cc 10475 ax-ac2 10503 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-map 8868 df-pm 8869 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-inf 9483 df-card 9979 df-acn 9982 df-ac 10156 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-n0 12527 df-z 12614 df-uz 12879 df-q 12991 df-rp 13035 df-ioo 13391 df-ico 13393 df-fl 13832 df-rest 17467 df-salg 46324 df-smblfn 46711 |
| This theorem is referenced by: smfpimgtxr 46795 smfpimgtmpt 46796 |
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