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Mirrors > Home > MPE Home > Th. List > blssm | Structured version Visualization version GIF version |
Description: A ball is a subset of the base set of a metric space. (Contributed by NM, 31-Aug-2006.) (Revised by Mario Carneiro, 12-Nov-2013.) |
Ref | Expression |
---|---|
blssm | ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑅) ⊆ 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | blf 23259 | . . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (ball‘𝐷):(𝑋 × ℝ*)⟶𝒫 𝑋) | |
2 | fovrn 7356 | . . 3 ⊢ (((ball‘𝐷):(𝑋 × ℝ*)⟶𝒫 𝑋 ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑅) ∈ 𝒫 𝑋) | |
3 | 1, 2 | syl3an1 1165 | . 2 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑅) ∈ 𝒫 𝑋) |
4 | 3 | elpwid 4510 | 1 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑅) ⊆ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1089 ∈ wcel 2112 ⊆ wss 3853 𝒫 cpw 4499 × cxp 5534 ⟶wf 6354 ‘cfv 6358 (class class class)co 7191 ℝ*cxr 10831 ∞Metcxmet 20302 ballcbl 20304 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-fv 6366 df-ov 7194 df-oprab 7195 df-mpo 7196 df-1st 7739 df-2nd 7740 df-map 8488 df-xr 10836 df-psmet 20309 df-xmet 20310 df-bl 20312 |
This theorem is referenced by: blpnfctr 23288 xmetresbl 23289 imasf1oxms 23341 prdsbl 23343 blcld 23357 blcls 23358 prdsxmslem2 23381 metcnp 23393 cnllycmp 23807 lebnumlem3 23814 lebnum 23815 cfil3i 24120 iscfil3 24124 cfilfcls 24125 iscmet3lem2 24143 equivcfil 24150 caublcls 24160 relcmpcmet 24169 cmpcmet 24170 cncmet 24173 bcthlem2 24176 bcthlem4 24178 dvlip2 24846 dv11cn 24852 pserdvlem2 25274 pserdv 25275 abelthlem3 25279 abelthlem5 25281 dvlog2lem 25494 dvlog2 25495 efopnlem2 25499 efopn 25500 logtayl 25502 efrlim 25806 blsconn 32873 sstotbnd2 35618 equivtotbnd 35622 isbnd2 35627 blbnd 35631 totbndbnd 35633 prdstotbnd 35638 prdsbnd2 35639 ismtyima 35647 heiborlem3 35657 heiborlem8 35662 |
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