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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 22425 | . . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (ball‘𝐷):(𝑋 × ℝ*)⟶𝒫 𝑋) | |
2 | fovrn 6949 | . . 3 ⊢ (((ball‘𝐷):(𝑋 × ℝ*)⟶𝒫 𝑋 ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑅) ∈ 𝒫 𝑋) | |
3 | 1, 2 | syl3an1 1166 | . 2 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑅) ∈ 𝒫 𝑋) |
4 | 3 | elpwid 4309 | 1 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑅) ⊆ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1071 ∈ wcel 2145 ⊆ wss 3723 𝒫 cpw 4297 × cxp 5247 ⟶wf 6025 ‘cfv 6029 (class class class)co 6791 ℝ*cxr 10273 ∞Metcxmt 19939 ballcbl 19941 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7094 ax-cnex 10192 ax-resscn 10193 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-op 4323 df-uni 4575 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-id 5157 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-iota 5992 df-fun 6031 df-fn 6032 df-f 6033 df-fv 6037 df-ov 6794 df-oprab 6795 df-mpt2 6796 df-1st 7313 df-2nd 7314 df-map 8009 df-xr 10278 df-psmet 19946 df-xmet 19947 df-bl 19949 |
This theorem is referenced by: blpnfctr 22454 xmetresbl 22455 imasf1oxms 22507 prdsbl 22509 blcld 22523 blcls 22524 prdsxmslem2 22547 metcnp 22559 cnllycmp 22968 lebnumlem3 22975 lebnum 22976 cfil3i 23279 iscfil3 23283 cfilfcls 23284 iscmet3lem2 23302 equivcfil 23309 caublcls 23319 relcmpcmet 23327 cmpcmet 23328 cncmet 23331 bcthlem2 23334 bcthlem4 23336 dvlip2 23971 dv11cn 23977 pserdvlem2 24395 pserdv 24396 abelthlem3 24400 abelthlem5 24402 dvlog2lem 24612 dvlog2 24613 efopnlem2 24617 efopn 24618 logtayl 24620 efrlim 24910 blsconn 31557 sstotbnd2 33898 equivtotbnd 33902 isbnd2 33907 blbnd 33911 totbndbnd 33913 prdstotbnd 33918 prdsbnd2 33919 ismtyima 33927 heiborlem3 33937 heiborlem8 33942 |
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