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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 23019 | . . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (ball‘𝐷):(𝑋 × ℝ*)⟶𝒫 𝑋) | |
2 | fovrn 7320 | . . 3 ⊢ (((ball‘𝐷):(𝑋 × ℝ*)⟶𝒫 𝑋 ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑅) ∈ 𝒫 𝑋) | |
3 | 1, 2 | syl3an1 1159 | . 2 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑅) ∈ 𝒫 𝑋) |
4 | 3 | elpwid 4552 | 1 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑅) ⊆ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 ∈ wcel 2114 ⊆ wss 3938 𝒫 cpw 4541 × cxp 5555 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 ℝ*cxr 10676 ∞Metcxmet 20532 ballcbl 20534 |
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 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-1st 7691 df-2nd 7692 df-map 8410 df-xr 10681 df-psmet 20539 df-xmet 20540 df-bl 20542 |
This theorem is referenced by: blpnfctr 23048 xmetresbl 23049 imasf1oxms 23101 prdsbl 23103 blcld 23117 blcls 23118 prdsxmslem2 23141 metcnp 23153 cnllycmp 23562 lebnumlem3 23569 lebnum 23570 cfil3i 23874 iscfil3 23878 cfilfcls 23879 iscmet3lem2 23897 equivcfil 23904 caublcls 23914 relcmpcmet 23923 cmpcmet 23924 cncmet 23927 bcthlem2 23930 bcthlem4 23932 dvlip2 24594 dv11cn 24600 pserdvlem2 25018 pserdv 25019 abelthlem3 25023 abelthlem5 25025 dvlog2lem 25237 dvlog2 25238 efopnlem2 25242 efopn 25243 logtayl 25245 efrlim 25549 blsconn 32493 sstotbnd2 35054 equivtotbnd 35058 isbnd2 35063 blbnd 35067 totbndbnd 35069 prdstotbnd 35074 prdsbnd2 35075 ismtyima 35083 heiborlem3 35093 heiborlem8 35098 |
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