Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cnfsmf | Structured version Visualization version GIF version |
Description: A continuous function is measurable. Proposition 121D (b) of [Fremlin1] p. 36 is a special case of this theorem, where the topology on the domain is induced by the standard topology on n-dimensional Real numbers. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
cnfsmf.1 | ⊢ (𝜑 → 𝐽 ∈ Top) |
cnfsmf.k | ⊢ 𝐾 = (topGen‘ran (,)) |
cnfsmf.f | ⊢ (𝜑 → 𝐹 ∈ ((𝐽 ↾t dom 𝐹) Cn 𝐾)) |
cnfsmf.s | ⊢ 𝑆 = (SalGen‘𝐽) |
Ref | Expression |
---|---|
cnfsmf | ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1915 | . 2 ⊢ Ⅎ𝑎𝜑 | |
2 | cnfsmf.1 | . . 3 ⊢ (𝜑 → 𝐽 ∈ Top) | |
3 | cnfsmf.s | . . 3 ⊢ 𝑆 = (SalGen‘𝐽) | |
4 | 2, 3 | salgencld 42722 | . 2 ⊢ (𝜑 → 𝑆 ∈ SAlg) |
5 | cnfsmf.f | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ ((𝐽 ↾t dom 𝐹) Cn 𝐾)) | |
6 | eqid 2821 | . . . . . 6 ⊢ ∪ (𝐽 ↾t dom 𝐹) = ∪ (𝐽 ↾t dom 𝐹) | |
7 | eqid 2821 | . . . . . 6 ⊢ ∪ 𝐾 = ∪ 𝐾 | |
8 | 6, 7 | cnf 21837 | . . . . 5 ⊢ (𝐹 ∈ ((𝐽 ↾t dom 𝐹) Cn 𝐾) → 𝐹:∪ (𝐽 ↾t dom 𝐹)⟶∪ 𝐾) |
9 | 5, 8 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐹:∪ (𝐽 ↾t dom 𝐹)⟶∪ 𝐾) |
10 | 9 | fdmd 6509 | . . 3 ⊢ (𝜑 → dom 𝐹 = ∪ (𝐽 ↾t dom 𝐹)) |
11 | ovex 7175 | . . . . . . . 8 ⊢ (𝐽 ↾t dom 𝐹) ∈ V | |
12 | 11 | uniex 7453 | . . . . . . 7 ⊢ ∪ (𝐽 ↾t dom 𝐹) ∈ V |
13 | 12 | a1i 11 | . . . . . 6 ⊢ (𝜑 → ∪ (𝐽 ↾t dom 𝐹) ∈ V) |
14 | 10, 13 | eqeltrd 2913 | . . . . 5 ⊢ (𝜑 → dom 𝐹 ∈ V) |
15 | 2, 14 | unirestss 41480 | . . . 4 ⊢ (𝜑 → ∪ (𝐽 ↾t dom 𝐹) ⊆ ∪ 𝐽) |
16 | 3 | sssalgen 42708 | . . . . . 6 ⊢ (𝐽 ∈ Top → 𝐽 ⊆ 𝑆) |
17 | 2, 16 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐽 ⊆ 𝑆) |
18 | 17 | unissd 4834 | . . . 4 ⊢ (𝜑 → ∪ 𝐽 ⊆ ∪ 𝑆) |
19 | 15, 18 | sstrd 3965 | . . 3 ⊢ (𝜑 → ∪ (𝐽 ↾t dom 𝐹) ⊆ ∪ 𝑆) |
20 | 10, 19 | eqsstrd 3993 | . 2 ⊢ (𝜑 → dom 𝐹 ⊆ ∪ 𝑆) |
21 | uniretop 23354 | . . . . . . 7 ⊢ ℝ = ∪ (topGen‘ran (,)) | |
22 | cnfsmf.k | . . . . . . . 8 ⊢ 𝐾 = (topGen‘ran (,)) | |
23 | 22 | unieqi 4837 | . . . . . . 7 ⊢ ∪ 𝐾 = ∪ (topGen‘ran (,)) |
24 | 21, 23 | eqtr4i 2847 | . . . . . 6 ⊢ ℝ = ∪ 𝐾 |
25 | 24 | a1i 11 | . . . . 5 ⊢ (𝜑 → ℝ = ∪ 𝐾) |
26 | 25 | feq3d 6487 | . . . 4 ⊢ (𝜑 → (𝐹:∪ (𝐽 ↾t dom 𝐹)⟶ℝ ↔ 𝐹:∪ (𝐽 ↾t dom 𝐹)⟶∪ 𝐾)) |
27 | 9, 26 | mpbird 259 | . . 3 ⊢ (𝜑 → 𝐹:∪ (𝐽 ↾t dom 𝐹)⟶ℝ) |
28 | 27 | ffdmd 6523 | . 2 ⊢ (𝜑 → 𝐹:dom 𝐹⟶ℝ) |
29 | ssrest 21767 | . . . . 5 ⊢ ((𝑆 ∈ SAlg ∧ 𝐽 ⊆ 𝑆) → (𝐽 ↾t dom 𝐹) ⊆ (𝑆 ↾t dom 𝐹)) | |
30 | 4, 17, 29 | syl2anc 586 | . . . 4 ⊢ (𝜑 → (𝐽 ↾t dom 𝐹) ⊆ (𝑆 ↾t dom 𝐹)) |
31 | 30 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → (𝐽 ↾t dom 𝐹) ⊆ (𝑆 ↾t dom 𝐹)) |
32 | 10 | rabeqdv 3476 | . . . . 5 ⊢ (𝜑 → {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = {𝑥 ∈ ∪ (𝐽 ↾t dom 𝐹) ∣ (𝐹‘𝑥) < 𝑎}) |
33 | 32 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = {𝑥 ∈ ∪ (𝐽 ↾t dom 𝐹) ∣ (𝐹‘𝑥) < 𝑎}) |
34 | nfcv 2977 | . . . . 5 ⊢ Ⅎ𝑥𝑎 | |
35 | nfcv 2977 | . . . . 5 ⊢ Ⅎ𝑥𝐹 | |
36 | nfv 1915 | . . . . 5 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑎 ∈ ℝ) | |
37 | eqid 2821 | . . . . 5 ⊢ {𝑥 ∈ ∪ (𝐽 ↾t dom 𝐹) ∣ (𝐹‘𝑥) < 𝑎} = {𝑥 ∈ ∪ (𝐽 ↾t dom 𝐹) ∣ (𝐹‘𝑥) < 𝑎} | |
38 | rexr 10673 | . . . . . 6 ⊢ (𝑎 ∈ ℝ → 𝑎 ∈ ℝ*) | |
39 | 38 | adantl 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝑎 ∈ ℝ*) |
40 | 5 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝐹 ∈ ((𝐽 ↾t dom 𝐹) Cn 𝐾)) |
41 | 34, 35, 36, 22, 6, 37, 39, 40 | rfcnpre2 41378 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ ∪ (𝐽 ↾t dom 𝐹) ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝐽 ↾t dom 𝐹)) |
42 | 33, 41 | eqeltrd 2913 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝐽 ↾t dom 𝐹)) |
43 | 31, 42 | sseldd 3956 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t dom 𝐹)) |
44 | 1, 4, 20, 28, 43 | issmfd 43102 | 1 ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 {crab 3142 Vcvv 3486 ⊆ wss 3924 ∪ cuni 4824 class class class wbr 5052 dom cdm 5541 ran crn 5542 ⟶wf 6337 ‘cfv 6341 (class class class)co 7142 ℝcr 10522 ℝ*cxr 10660 < clt 10661 (,)cioo 12725 ↾t crest 16677 topGenctg 16694 Topctop 21484 Cn ccn 21815 SAlgcsalg 42683 SalGencsalgen 42687 SMblFncsmblfn 43067 |
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 2793 ax-rep 5176 ax-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447 ax-cnex 10579 ax-resscn 10580 ax-1cn 10581 ax-icn 10582 ax-addcl 10583 ax-addrcl 10584 ax-mulcl 10585 ax-mulrcl 10586 ax-mulcom 10587 ax-addass 10588 ax-mulass 10589 ax-distr 10590 ax-i2m1 10591 ax-1ne0 10592 ax-1rid 10593 ax-rnegex 10594 ax-rrecex 10595 ax-cnre 10596 ax-pre-lttri 10597 ax-pre-lttrn 10598 ax-pre-ltadd 10599 ax-pre-mulgt0 10600 ax-pre-sup 10601 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3488 df-sbc 3764 df-csb 3872 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-pss 3942 df-nul 4280 df-if 4454 df-pw 4527 df-sn 4554 df-pr 4556 df-tp 4558 df-op 4560 df-uni 4825 df-int 4863 df-iun 4907 df-br 5053 df-opab 5115 df-mpt 5133 df-tr 5159 df-id 5446 df-eprel 5451 df-po 5460 df-so 5461 df-fr 5500 df-we 5502 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-res 5553 df-ima 5554 df-pred 6134 df-ord 6180 df-on 6181 df-lim 6182 df-suc 6183 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-f1 6346 df-fo 6347 df-f1o 6348 df-fv 6349 df-riota 7100 df-ov 7145 df-oprab 7146 df-mpo 7147 df-om 7567 df-1st 7675 df-2nd 7676 df-wrecs 7933 df-recs 7994 df-rdg 8032 df-er 8275 df-map 8394 df-pm 8395 df-en 8496 df-dom 8497 df-sdom 8498 df-sup 8892 df-inf 8893 df-pnf 10663 df-mnf 10664 df-xr 10665 df-ltxr 10666 df-le 10667 df-sub 10858 df-neg 10859 df-div 11284 df-nn 11625 df-n0 11885 df-z 11969 df-uz 12231 df-q 12336 df-ioo 12729 df-ico 12731 df-rest 16679 df-topgen 16700 df-top 21485 df-topon 21502 df-bases 21537 df-cn 21818 df-salg 42684 df-salgen 42688 df-smblfn 43068 |
This theorem is referenced by: cnfrrnsmf 43118 |
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