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Mirrors > Home > MPE Home > Th. List > mopni2 | Structured version Visualization version GIF version |
Description: An open set of a metric space includes a ball around each of its points. (Contributed by NM, 2-May-2007.) (Revised by Mario Carneiro, 12-Nov-2013.) |
Ref | Expression |
---|---|
mopni.1 | ⊢ 𝐽 = (MetOpen‘𝐷) |
Ref | Expression |
---|---|
mopni2 | ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝐽 ∧ 𝑃 ∈ 𝐴) → ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mopni.1 | . . 3 ⊢ 𝐽 = (MetOpen‘𝐷) | |
2 | 1 | mopni 23344 | . 2 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝐽 ∧ 𝑃 ∈ 𝐴) → ∃𝑦 ∈ ran (ball‘𝐷)(𝑃 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴)) |
3 | 1 | mopnss 23298 | . . . . 5 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝐽) → 𝐴 ⊆ 𝑋) |
4 | 3 | sselda 3887 | . . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝐽) ∧ 𝑃 ∈ 𝐴) → 𝑃 ∈ 𝑋) |
5 | blssex 23279 | . . . . 5 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) → (∃𝑦 ∈ ran (ball‘𝐷)(𝑃 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴) ↔ ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ 𝐴)) | |
6 | 5 | adantlr 715 | . . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝐽) ∧ 𝑃 ∈ 𝑋) → (∃𝑦 ∈ ran (ball‘𝐷)(𝑃 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴) ↔ ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ 𝐴)) |
7 | 4, 6 | syldan 594 | . . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝐽) ∧ 𝑃 ∈ 𝐴) → (∃𝑦 ∈ ran (ball‘𝐷)(𝑃 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴) ↔ ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ 𝐴)) |
8 | 7 | 3impa 1112 | . 2 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝐽 ∧ 𝑃 ∈ 𝐴) → (∃𝑦 ∈ ran (ball‘𝐷)(𝑃 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴) ↔ ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ 𝐴)) |
9 | 2, 8 | mpbid 235 | 1 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝐽 ∧ 𝑃 ∈ 𝐴) → ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2112 ∃wrex 3052 ⊆ wss 3853 ran crn 5537 ‘cfv 6358 (class class class)co 7191 ℝ+crp 12551 ∞Metcxmet 20302 ballcbl 20304 MetOpencmopn 20307 |
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 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 ax-pre-sup 10772 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 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-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 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-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 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-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-er 8369 df-map 8488 df-en 8605 df-dom 8606 df-sdom 8607 df-sup 9036 df-inf 9037 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-div 11455 df-nn 11796 df-2 11858 df-n0 12056 df-z 12142 df-uz 12404 df-q 12510 df-rp 12552 df-xneg 12669 df-xadd 12670 df-xmul 12671 df-topgen 16902 df-psmet 20309 df-xmet 20310 df-bl 20312 df-mopn 20313 df-top 21745 df-topon 21762 df-bases 21797 |
This theorem is referenced by: mopni3 23346 neibl 23353 met1stc 23373 met2ndci 23374 prdsxmslem2 23381 metcnp3 23392 xrsmopn 23663 iccntr 23672 icccmplem3 23675 reconnlem2 23678 opnreen 23682 metdseq0 23705 cnllycmp 23807 nmhmcn 23971 lmmbr 24109 cfilfcls 24125 iscmet3lem2 24143 bcthlem5 24179 opnmbllem 24452 ellimc3 24730 lhop 24867 dvcnvre 24870 xrlimcnp 25805 lgamucov 25874 ubthlem1 28905 cnllysconn 32874 ptrecube 35463 opnmbllem0 35499 heiborlem8 35662 qndenserrnopnlem 43456 opnvonmbllem2 43789 |
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