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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > mbfpsssmf | Structured version Visualization version GIF version |
Description: Real-valued measurable functions are a proper subset of sigma-measurable functions (w.r.t. the Lebesgue measure on the reals). (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
mbfpsssmf.1 | ⊢ 𝑆 = dom vol |
Ref | Expression |
---|---|
mbfpsssmf | ⊢ (MblFn ∩ (ℝ ↑pm ℝ)) ⊊ (SMblFn‘𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elinel1 4156 | . . . . 5 ⊢ (𝑓 ∈ (MblFn ∩ (ℝ ↑pm ℝ)) → 𝑓 ∈ MblFn) | |
2 | elinel2 4157 | . . . . . 6 ⊢ (𝑓 ∈ (MblFn ∩ (ℝ ↑pm ℝ)) → 𝑓 ∈ (ℝ ↑pm ℝ)) | |
3 | elpmrn 43528 | . . . . . 6 ⊢ (𝑓 ∈ (ℝ ↑pm ℝ) → ran 𝑓 ⊆ ℝ) | |
4 | 2, 3 | syl 17 | . . . . 5 ⊢ (𝑓 ∈ (MblFn ∩ (ℝ ↑pm ℝ)) → ran 𝑓 ⊆ ℝ) |
5 | mbfpsssmf.1 | . . . . 5 ⊢ 𝑆 = dom vol | |
6 | 1, 4, 5 | mbfresmf 45066 | . . . 4 ⊢ (𝑓 ∈ (MblFn ∩ (ℝ ↑pm ℝ)) → 𝑓 ∈ (SMblFn‘𝑆)) |
7 | 6 | ssriv 3949 | . . 3 ⊢ (MblFn ∩ (ℝ ↑pm ℝ)) ⊆ (SMblFn‘𝑆) |
8 | 5 | nsssmfmbf 45106 | . . . 4 ⊢ ¬ (SMblFn‘𝑆) ⊆ MblFn |
9 | 1 | ssriv 3949 | . . . 4 ⊢ (MblFn ∩ (ℝ ↑pm ℝ)) ⊆ MblFn |
10 | nsstr 43393 | . . . 4 ⊢ ((¬ (SMblFn‘𝑆) ⊆ MblFn ∧ (MblFn ∩ (ℝ ↑pm ℝ)) ⊆ MblFn) → ¬ (SMblFn‘𝑆) ⊆ (MblFn ∩ (ℝ ↑pm ℝ))) | |
11 | 8, 9, 10 | mp2an 691 | . . 3 ⊢ ¬ (SMblFn‘𝑆) ⊆ (MblFn ∩ (ℝ ↑pm ℝ)) |
12 | 7, 11 | pm3.2i 472 | . 2 ⊢ ((MblFn ∩ (ℝ ↑pm ℝ)) ⊆ (SMblFn‘𝑆) ∧ ¬ (SMblFn‘𝑆) ⊆ (MblFn ∩ (ℝ ↑pm ℝ))) |
13 | dfpss3 4047 | . 2 ⊢ ((MblFn ∩ (ℝ ↑pm ℝ)) ⊊ (SMblFn‘𝑆) ↔ ((MblFn ∩ (ℝ ↑pm ℝ)) ⊆ (SMblFn‘𝑆) ∧ ¬ (SMblFn‘𝑆) ⊆ (MblFn ∩ (ℝ ↑pm ℝ)))) | |
14 | 12, 13 | mpbir 230 | 1 ⊢ (MblFn ∩ (ℝ ↑pm ℝ)) ⊊ (SMblFn‘𝑆) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∩ cin 3910 ⊆ wss 3911 ⊊ wpss 3912 dom cdm 5634 ran crn 5635 ‘cfv 6497 (class class class)co 7358 ↑pm cpm 8769 ℝcr 11055 volcvol 24843 MblFncmbf 24994 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-disj 5072 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-of 7618 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-2o 8414 df-oadd 8417 df-omul 8418 df-er 8651 df-ec 8653 df-qs 8657 df-map 8770 df-pm 8771 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-fi 9352 df-sup 9383 df-inf 9384 df-oi 9451 df-dju 9842 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-2 12221 df-3 12222 df-n0 12419 df-z 12505 df-uz 12769 df-q 12879 df-rp 12921 df-xneg 13038 df-xadd 13039 df-xmul 13040 df-ioo 13274 df-ico 13276 df-icc 13277 df-fz 13431 df-fzo 13574 df-fl 13703 df-seq 13913 df-exp 13974 df-hash 14237 df-cj 14990 df-re 14991 df-im 14992 df-sqrt 15126 df-abs 15127 df-clim 15376 df-rlim 15377 df-sum 15577 df-rest 17309 df-topgen 17330 df-psmet 20804 df-xmet 20805 df-met 20806 df-bl 20807 df-mopn 20808 df-top 22259 df-topon 22276 df-bases 22312 df-cmp 22754 df-ovol 24844 df-vol 24845 df-mbf 24999 df-salg 44636 df-smblfn 45023 |
This theorem is referenced by: (None) |
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