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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 42989 | . 2 ⊢ (𝜑 → 𝑆 ∈ SAlg) |
5 | cnfsmf.f | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ ((𝐽 ↾t dom 𝐹) Cn 𝐾)) | |
6 | eqid 2798 | . . . . . 6 ⊢ ∪ (𝐽 ↾t dom 𝐹) = ∪ (𝐽 ↾t dom 𝐹) | |
7 | eqid 2798 | . . . . . 6 ⊢ ∪ 𝐾 = ∪ 𝐾 | |
8 | 6, 7 | cnf 21851 | . . . . 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 6497 | . . 3 ⊢ (𝜑 → dom 𝐹 = ∪ (𝐽 ↾t dom 𝐹)) |
11 | ovex 7168 | . . . . . . . 8 ⊢ (𝐽 ↾t dom 𝐹) ∈ V | |
12 | 11 | uniex 7447 | . . . . . . 7 ⊢ ∪ (𝐽 ↾t dom 𝐹) ∈ V |
13 | 12 | a1i 11 | . . . . . 6 ⊢ (𝜑 → ∪ (𝐽 ↾t dom 𝐹) ∈ V) |
14 | 10, 13 | eqeltrd 2890 | . . . . 5 ⊢ (𝜑 → dom 𝐹 ∈ V) |
15 | 2, 14 | unirestss 41759 | . . . 4 ⊢ (𝜑 → ∪ (𝐽 ↾t dom 𝐹) ⊆ ∪ 𝐽) |
16 | 3 | sssalgen 42975 | . . . . . 6 ⊢ (𝐽 ∈ Top → 𝐽 ⊆ 𝑆) |
17 | 2, 16 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐽 ⊆ 𝑆) |
18 | 17 | unissd 4810 | . . . 4 ⊢ (𝜑 → ∪ 𝐽 ⊆ ∪ 𝑆) |
19 | 15, 18 | sstrd 3925 | . . 3 ⊢ (𝜑 → ∪ (𝐽 ↾t dom 𝐹) ⊆ ∪ 𝑆) |
20 | 10, 19 | eqsstrd 3953 | . 2 ⊢ (𝜑 → dom 𝐹 ⊆ ∪ 𝑆) |
21 | uniretop 23368 | . . . . . . 7 ⊢ ℝ = ∪ (topGen‘ran (,)) | |
22 | cnfsmf.k | . . . . . . . 8 ⊢ 𝐾 = (topGen‘ran (,)) | |
23 | 22 | unieqi 4813 | . . . . . . 7 ⊢ ∪ 𝐾 = ∪ (topGen‘ran (,)) |
24 | 21, 23 | eqtr4i 2824 | . . . . . 6 ⊢ ℝ = ∪ 𝐾 |
25 | 24 | a1i 11 | . . . . 5 ⊢ (𝜑 → ℝ = ∪ 𝐾) |
26 | 25 | feq3d 6474 | . . . 4 ⊢ (𝜑 → (𝐹:∪ (𝐽 ↾t dom 𝐹)⟶ℝ ↔ 𝐹:∪ (𝐽 ↾t dom 𝐹)⟶∪ 𝐾)) |
27 | 9, 26 | mpbird 260 | . . 3 ⊢ (𝜑 → 𝐹:∪ (𝐽 ↾t dom 𝐹)⟶ℝ) |
28 | 27 | ffdmd 6511 | . 2 ⊢ (𝜑 → 𝐹:dom 𝐹⟶ℝ) |
29 | ssrest 21781 | . . . . 5 ⊢ ((𝑆 ∈ SAlg ∧ 𝐽 ⊆ 𝑆) → (𝐽 ↾t dom 𝐹) ⊆ (𝑆 ↾t dom 𝐹)) | |
30 | 4, 17, 29 | syl2anc 587 | . . . 4 ⊢ (𝜑 → (𝐽 ↾t dom 𝐹) ⊆ (𝑆 ↾t dom 𝐹)) |
31 | 30 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → (𝐽 ↾t dom 𝐹) ⊆ (𝑆 ↾t dom 𝐹)) |
32 | 10 | rabeqdv 3432 | . . . . 5 ⊢ (𝜑 → {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = {𝑥 ∈ ∪ (𝐽 ↾t dom 𝐹) ∣ (𝐹‘𝑥) < 𝑎}) |
33 | 32 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = {𝑥 ∈ ∪ (𝐽 ↾t dom 𝐹) ∣ (𝐹‘𝑥) < 𝑎}) |
34 | nfcv 2955 | . . . . 5 ⊢ Ⅎ𝑥𝑎 | |
35 | nfcv 2955 | . . . . 5 ⊢ Ⅎ𝑥𝐹 | |
36 | nfv 1915 | . . . . 5 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑎 ∈ ℝ) | |
37 | eqid 2798 | . . . . 5 ⊢ {𝑥 ∈ ∪ (𝐽 ↾t dom 𝐹) ∣ (𝐹‘𝑥) < 𝑎} = {𝑥 ∈ ∪ (𝐽 ↾t dom 𝐹) ∣ (𝐹‘𝑥) < 𝑎} | |
38 | rexr 10676 | . . . . . 6 ⊢ (𝑎 ∈ ℝ → 𝑎 ∈ ℝ*) | |
39 | 38 | adantl 485 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝑎 ∈ ℝ*) |
40 | 5 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝐹 ∈ ((𝐽 ↾t dom 𝐹) Cn 𝐾)) |
41 | 34, 35, 36, 22, 6, 37, 39, 40 | rfcnpre2 41660 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ ∪ (𝐽 ↾t dom 𝐹) ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝐽 ↾t dom 𝐹)) |
42 | 33, 41 | eqeltrd 2890 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝐽 ↾t dom 𝐹)) |
43 | 31, 42 | sseldd 3916 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t dom 𝐹)) |
44 | 1, 4, 20, 28, 43 | issmfd 43369 | 1 ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 {crab 3110 Vcvv 3441 ⊆ wss 3881 ∪ cuni 4800 class class class wbr 5030 dom cdm 5519 ran crn 5520 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 ℝcr 10525 ℝ*cxr 10663 < clt 10664 (,)cioo 12726 ↾t crest 16686 topGenctg 16703 Topctop 21498 Cn ccn 21829 SAlgcsalg 42950 SalGencsalgen 42954 SMblFncsmblfn 43334 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-map 8391 df-pm 8392 df-en 8493 df-dom 8494 df-sdom 8495 df-sup 8890 df-inf 8891 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-n0 11886 df-z 11970 df-uz 12232 df-q 12337 df-ioo 12730 df-ico 12732 df-rest 16688 df-topgen 16709 df-top 21499 df-topon 21516 df-bases 21551 df-cn 21832 df-salg 42951 df-salgen 42955 df-smblfn 43335 |
This theorem is referenced by: cnfrrnsmf 43385 |
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