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Mathbox for Glauco Siliprandi |
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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 5907 | . . . . 5 ⊢ Ⅎ𝑥dom 𝐹 |
4 | 1, 3 | nfcxfr 2902 | . . . 4 ⊢ Ⅎ𝑥𝐷 |
5 | nfcv 2904 | . . . 4 ⊢ Ⅎ𝑦𝐷 | |
6 | nfv 1918 | . . . 4 ⊢ Ⅎ𝑦 𝐴 < (𝐹‘𝑥) | |
7 | nfcv 2904 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
8 | nfcv 2904 | . . . . 5 ⊢ Ⅎ𝑥 < | |
9 | nfcv 2904 | . . . . . 6 ⊢ Ⅎ𝑥𝑦 | |
10 | 2, 9 | nffv 6853 | . . . . 5 ⊢ Ⅎ𝑥(𝐹‘𝑦) |
11 | 7, 8, 10 | nfbr 5153 | . . . 4 ⊢ Ⅎ𝑥 𝐴 < (𝐹‘𝑦) |
12 | fveq2 6843 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦)) | |
13 | 12 | breq2d 5118 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝐴 < (𝐹‘𝑥) ↔ 𝐴 < (𝐹‘𝑦))) |
14 | 4, 5, 6, 11, 13 | cbvrabw 3438 | . . 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 45089 | . 2 ⊢ (𝜑 → {𝑦 ∈ 𝐷 ∣ 𝐴 < (𝐹‘𝑦)} ∈ (𝑆 ↾t 𝐷)) |
20 | 15, 19 | eqeltrd 2834 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝐷 ∣ 𝐴 < (𝐹‘𝑥)} ∈ (𝑆 ↾t 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 Ⅎwnfc 2884 {crab 3406 class class class wbr 5106 dom cdm 5634 ‘cfv 6497 (class class class)co 7358 ℝcr 11055 < clt 11194 ↾t crest 17307 SAlgcsalg 44635 SMblFncsmblfn 45022 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-inf2 9582 ax-cc 10376 ax-ac2 10404 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 ax-pre-sup 11134 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-iin 4958 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-se 5590 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-isom 6506 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8651 df-map 8770 df-pm 8771 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-sup 9383 df-inf 9384 df-card 9880 df-acn 9883 df-ac 10057 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-div 11818 df-nn 12159 df-n0 12419 df-z 12505 df-uz 12769 df-q 12879 df-rp 12921 df-ioo 13274 df-ico 13276 df-fl 13703 df-rest 17309 df-salg 44636 df-smblfn 45023 |
This theorem is referenced by: smfpimgtxr 45107 smfpimgtmpt 45108 |
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