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Mathbox for Jeff Madsen |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > totbndss | Structured version Visualization version GIF version |
Description: A subset of a totally bounded metric space is totally bounded. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 12-Sep-2015.) |
Ref | Expression |
---|---|
totbndss | ⊢ ((𝑀 ∈ (TotBnd‘𝑋) ∧ 𝑆 ⊆ 𝑋) → (𝑀 ↾ (𝑆 × 𝑆)) ∈ (TotBnd‘𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | istotbnd 34055 | . . . 4 ⊢ (𝑀 ∈ (TotBnd‘𝑋) ↔ (𝑀 ∈ (Met‘𝑋) ∧ ∀𝑑 ∈ ℝ+ ∃𝑣 ∈ Fin (∪ 𝑣 = 𝑋 ∧ ∀𝑏 ∈ 𝑣 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)))) | |
2 | 1 | simprbi 491 | . . 3 ⊢ (𝑀 ∈ (TotBnd‘𝑋) → ∀𝑑 ∈ ℝ+ ∃𝑣 ∈ Fin (∪ 𝑣 = 𝑋 ∧ ∀𝑏 ∈ 𝑣 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑))) |
3 | sseq2 3823 | . . . . . . 7 ⊢ (∪ 𝑣 = 𝑋 → (𝑆 ⊆ ∪ 𝑣 ↔ 𝑆 ⊆ 𝑋)) | |
4 | 3 | biimprcd 242 | . . . . . 6 ⊢ (𝑆 ⊆ 𝑋 → (∪ 𝑣 = 𝑋 → 𝑆 ⊆ ∪ 𝑣)) |
5 | 4 | anim1d 605 | . . . . 5 ⊢ (𝑆 ⊆ 𝑋 → ((∪ 𝑣 = 𝑋 ∧ ∀𝑏 ∈ 𝑣 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)) → (𝑆 ⊆ ∪ 𝑣 ∧ ∀𝑏 ∈ 𝑣 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)))) |
6 | 5 | reximdv 3196 | . . . 4 ⊢ (𝑆 ⊆ 𝑋 → (∃𝑣 ∈ Fin (∪ 𝑣 = 𝑋 ∧ ∀𝑏 ∈ 𝑣 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)) → ∃𝑣 ∈ Fin (𝑆 ⊆ ∪ 𝑣 ∧ ∀𝑏 ∈ 𝑣 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)))) |
7 | 6 | ralimdv 3144 | . . 3 ⊢ (𝑆 ⊆ 𝑋 → (∀𝑑 ∈ ℝ+ ∃𝑣 ∈ Fin (∪ 𝑣 = 𝑋 ∧ ∀𝑏 ∈ 𝑣 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)) → ∀𝑑 ∈ ℝ+ ∃𝑣 ∈ Fin (𝑆 ⊆ ∪ 𝑣 ∧ ∀𝑏 ∈ 𝑣 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)))) |
8 | 2, 7 | mpan9 503 | . 2 ⊢ ((𝑀 ∈ (TotBnd‘𝑋) ∧ 𝑆 ⊆ 𝑋) → ∀𝑑 ∈ ℝ+ ∃𝑣 ∈ Fin (𝑆 ⊆ ∪ 𝑣 ∧ ∀𝑏 ∈ 𝑣 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑))) |
9 | totbndmet 34058 | . . 3 ⊢ (𝑀 ∈ (TotBnd‘𝑋) → 𝑀 ∈ (Met‘𝑋)) | |
10 | eqid 2799 | . . . 4 ⊢ (𝑀 ↾ (𝑆 × 𝑆)) = (𝑀 ↾ (𝑆 × 𝑆)) | |
11 | 10 | sstotbnd 34061 | . . 3 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) → ((𝑀 ↾ (𝑆 × 𝑆)) ∈ (TotBnd‘𝑆) ↔ ∀𝑑 ∈ ℝ+ ∃𝑣 ∈ Fin (𝑆 ⊆ ∪ 𝑣 ∧ ∀𝑏 ∈ 𝑣 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)))) |
12 | 9, 11 | sylan 576 | . 2 ⊢ ((𝑀 ∈ (TotBnd‘𝑋) ∧ 𝑆 ⊆ 𝑋) → ((𝑀 ↾ (𝑆 × 𝑆)) ∈ (TotBnd‘𝑆) ↔ ∀𝑑 ∈ ℝ+ ∃𝑣 ∈ Fin (𝑆 ⊆ ∪ 𝑣 ∧ ∀𝑏 ∈ 𝑣 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)))) |
13 | 8, 12 | mpbird 249 | 1 ⊢ ((𝑀 ∈ (TotBnd‘𝑋) ∧ 𝑆 ⊆ 𝑋) → (𝑀 ↾ (𝑆 × 𝑆)) ∈ (TotBnd‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 385 = wceq 1653 ∈ wcel 2157 ∀wral 3089 ∃wrex 3090 ⊆ wss 3769 ∪ cuni 4628 × cxp 5310 ↾ cres 5314 ‘cfv 6101 (class class class)co 6878 Fincfn 8195 ℝ+crp 12074 Metcmet 20054 ballcbl 20055 TotBndctotbnd 34052 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-cnex 10280 ax-resscn 10281 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-addrcl 10285 ax-mulcl 10286 ax-mulrcl 10287 ax-mulcom 10288 ax-addass 10289 ax-mulass 10290 ax-distr 10291 ax-i2m1 10292 ax-1ne0 10293 ax-1rid 10294 ax-rnegex 10295 ax-rrecex 10296 ax-cnre 10297 ax-pre-lttri 10298 ax-pre-lttrn 10299 ax-pre-ltadd 10300 ax-pre-mulgt0 10301 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-int 4668 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-om 7300 df-1st 7401 df-2nd 7402 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-1o 7799 df-oadd 7803 df-er 7982 df-map 8097 df-en 8196 df-dom 8197 df-sdom 8198 df-fin 8199 df-pnf 10365 df-mnf 10366 df-xr 10367 df-ltxr 10368 df-le 10369 df-sub 10558 df-neg 10559 df-div 10977 df-2 11376 df-rp 12075 df-xneg 12193 df-xadd 12194 df-xmul 12195 df-psmet 20060 df-xmet 20061 df-met 20062 df-bl 20063 df-totbnd 34054 |
This theorem is referenced by: prdsbnd2 34081 |
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