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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 24417 | . . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (ball‘𝐷):(𝑋 × ℝ*)⟶𝒫 𝑋) | |
| 2 | fovcdm 7603 | . . 3 ⊢ (((ball‘𝐷):(𝑋 × ℝ*)⟶𝒫 𝑋 ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑅) ∈ 𝒫 𝑋) | |
| 3 | 1, 2 | syl3an1 1164 | . 2 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑅) ∈ 𝒫 𝑋) |
| 4 | 3 | elpwid 4609 | 1 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑅) ⊆ 𝑋) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ w3a 1087 ∈ wcel 2108 ⊆ wss 3951 𝒫 cpw 4600 × cxp 5683 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ℝ*cxr 11294 ∞Metcxmet 21349 ballcbl 21351 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-map 8868 df-xr 11299 df-psmet 21356 df-xmet 21357 df-bl 21359 |
| This theorem is referenced by: blpnfctr 24446 xmetresbl 24447 imasf1oxms 24502 prdsbl 24504 blcld 24518 blcls 24519 prdsxmslem2 24542 metcnp 24554 cnllycmp 24988 lebnumlem3 24995 lebnum 24996 cfil3i 25303 iscfil3 25307 cfilfcls 25308 iscmet3lem2 25326 equivcfil 25333 caublcls 25343 relcmpcmet 25352 cmpcmet 25353 cncmet 25356 bcthlem2 25359 bcthlem4 25361 dvlip2 26034 dv11cn 26040 pserdvlem2 26472 pserdv 26473 abelthlem3 26477 abelthlem5 26479 dvlog2lem 26694 dvlog2 26695 efopnlem2 26699 efopn 26700 logtayl 26702 efrlim 27012 efrlimOLD 27013 blsconn 35249 sstotbnd2 37781 equivtotbnd 37785 isbnd2 37790 blbnd 37794 totbndbnd 37796 prdstotbnd 37801 prdsbnd2 37802 ismtyima 37810 heiborlem3 37820 heiborlem8 37825 |
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