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 1921 | . 2 ⊢ Ⅎ𝑎𝜑 | |
2 | cnfsmf.1 | . . 3 ⊢ (𝜑 → 𝐽 ∈ Top) | |
3 | cnfsmf.s | . . 3 ⊢ 𝑆 = (SalGen‘𝐽) | |
4 | 2, 3 | salgencld 43470 | . 2 ⊢ (𝜑 → 𝑆 ∈ SAlg) |
5 | cnfsmf.f | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ ((𝐽 ↾t dom 𝐹) Cn 𝐾)) | |
6 | eqid 2739 | . . . . . 6 ⊢ ∪ (𝐽 ↾t dom 𝐹) = ∪ (𝐽 ↾t dom 𝐹) | |
7 | eqid 2739 | . . . . . 6 ⊢ ∪ 𝐾 = ∪ 𝐾 | |
8 | 6, 7 | cnf 22009 | . . . . 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 6525 | . . 3 ⊢ (𝜑 → dom 𝐹 = ∪ (𝐽 ↾t dom 𝐹)) |
11 | ovex 7215 | . . . . . . . 8 ⊢ (𝐽 ↾t dom 𝐹) ∈ V | |
12 | 11 | uniex 7497 | . . . . . . 7 ⊢ ∪ (𝐽 ↾t dom 𝐹) ∈ V |
13 | 12 | a1i 11 | . . . . . 6 ⊢ (𝜑 → ∪ (𝐽 ↾t dom 𝐹) ∈ V) |
14 | 10, 13 | eqeltrd 2834 | . . . . 5 ⊢ (𝜑 → dom 𝐹 ∈ V) |
15 | 2, 14 | unirestss 42251 | . . . 4 ⊢ (𝜑 → ∪ (𝐽 ↾t dom 𝐹) ⊆ ∪ 𝐽) |
16 | 3 | sssalgen 43456 | . . . . . 6 ⊢ (𝐽 ∈ Top → 𝐽 ⊆ 𝑆) |
17 | 2, 16 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐽 ⊆ 𝑆) |
18 | 17 | unissd 4816 | . . . 4 ⊢ (𝜑 → ∪ 𝐽 ⊆ ∪ 𝑆) |
19 | 15, 18 | sstrd 3897 | . . 3 ⊢ (𝜑 → ∪ (𝐽 ↾t dom 𝐹) ⊆ ∪ 𝑆) |
20 | 10, 19 | eqsstrd 3925 | . 2 ⊢ (𝜑 → dom 𝐹 ⊆ ∪ 𝑆) |
21 | uniretop 23527 | . . . . . . 7 ⊢ ℝ = ∪ (topGen‘ran (,)) | |
22 | cnfsmf.k | . . . . . . . 8 ⊢ 𝐾 = (topGen‘ran (,)) | |
23 | 22 | unieqi 4819 | . . . . . . 7 ⊢ ∪ 𝐾 = ∪ (topGen‘ran (,)) |
24 | 21, 23 | eqtr4i 2765 | . . . . . 6 ⊢ ℝ = ∪ 𝐾 |
25 | 24 | a1i 11 | . . . . 5 ⊢ (𝜑 → ℝ = ∪ 𝐾) |
26 | 25 | feq3d 6501 | . . . 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 6545 | . 2 ⊢ (𝜑 → 𝐹:dom 𝐹⟶ℝ) |
29 | ssrest 21939 | . . . . 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 3387 | . . . . 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 2900 | . . . . 5 ⊢ Ⅎ𝑥𝑎 | |
35 | nfcv 2900 | . . . . 5 ⊢ Ⅎ𝑥𝐹 | |
36 | nfv 1921 | . . . . 5 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑎 ∈ ℝ) | |
37 | eqid 2739 | . . . . 5 ⊢ {𝑥 ∈ ∪ (𝐽 ↾t dom 𝐹) ∣ (𝐹‘𝑥) < 𝑎} = {𝑥 ∈ ∪ (𝐽 ↾t dom 𝐹) ∣ (𝐹‘𝑥) < 𝑎} | |
38 | rexr 10777 | . . . . . 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 42152 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ ∪ (𝐽 ↾t dom 𝐹) ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝐽 ↾t dom 𝐹)) |
42 | 33, 41 | eqeltrd 2834 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝐽 ↾t dom 𝐹)) |
43 | 31, 42 | sseldd 3888 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t dom 𝐹)) |
44 | 1, 4, 20, 28, 43 | issmfd 43850 | 1 ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 {crab 3058 Vcvv 3400 ⊆ wss 3853 ∪ cuni 4806 class class class wbr 5040 dom cdm 5535 ran crn 5536 ⟶wf 6345 ‘cfv 6349 (class class class)co 7182 ℝcr 10626 ℝ*cxr 10764 < clt 10765 (,)cioo 12833 ↾t crest 16809 topGenctg 16826 Topctop 21656 Cn ccn 21987 SAlgcsalg 43431 SalGencsalgen 43435 SMblFncsmblfn 43815 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7491 ax-cnex 10683 ax-resscn 10684 ax-1cn 10685 ax-icn 10686 ax-addcl 10687 ax-addrcl 10688 ax-mulcl 10689 ax-mulrcl 10690 ax-mulcom 10691 ax-addass 10692 ax-mulass 10693 ax-distr 10694 ax-i2m1 10695 ax-1ne0 10696 ax-1rid 10697 ax-rnegex 10698 ax-rrecex 10699 ax-cnre 10700 ax-pre-lttri 10701 ax-pre-lttrn 10702 ax-pre-ltadd 10703 ax-pre-mulgt0 10704 ax-pre-sup 10705 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-tp 4531 df-op 4533 df-uni 4807 df-int 4847 df-iun 4893 df-br 5041 df-opab 5103 df-mpt 5121 df-tr 5147 df-id 5439 df-eprel 5444 df-po 5452 df-so 5453 df-fr 5493 df-we 5495 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-pred 6139 df-ord 6185 df-on 6186 df-lim 6187 df-suc 6188 df-iota 6307 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7139 df-ov 7185 df-oprab 7186 df-mpo 7187 df-om 7612 df-1st 7726 df-2nd 7727 df-wrecs 7988 df-recs 8049 df-rdg 8087 df-er 8332 df-map 8451 df-pm 8452 df-en 8568 df-dom 8569 df-sdom 8570 df-sup 8991 df-inf 8992 df-pnf 10767 df-mnf 10768 df-xr 10769 df-ltxr 10770 df-le 10771 df-sub 10962 df-neg 10963 df-div 11388 df-nn 11729 df-n0 11989 df-z 12075 df-uz 12337 df-q 12443 df-ioo 12837 df-ico 12839 df-rest 16811 df-topgen 16832 df-top 21657 df-topon 21674 df-bases 21709 df-cn 21990 df-salg 43432 df-salgen 43436 df-smblfn 43816 |
This theorem is referenced by: cnfrrnsmf 43866 |
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