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Mirrors > Home > MPE Home > Th. List > elmopn2 | Structured version Visualization version GIF version |
Description: A defining property of an open set of a metric space. (Contributed by NM, 5-May-2007.) (Revised by Mario Carneiro, 12-Nov-2013.) |
Ref | Expression |
---|---|
mopnval.1 | ⊢ 𝐽 = (MetOpen‘𝐷) |
Ref | Expression |
---|---|
elmopn2 | ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐴 ∈ 𝐽 ↔ (𝐴 ⊆ 𝑋 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ ℝ+ (𝑥(ball‘𝐷)𝑦) ⊆ 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mopnval.1 | . . 3 ⊢ 𝐽 = (MetOpen‘𝐷) | |
2 | 1 | elmopn 22624 | . 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐴 ∈ 𝐽 ↔ (𝐴 ⊆ 𝑋 ∧ ∀𝑥 ∈ 𝐴 ∃𝑧 ∈ ran (ball‘𝐷)(𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝐴)))) |
3 | ssel2 3822 | . . . . . 6 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝑋) | |
4 | blssex 22609 | . . . . . 6 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) → (∃𝑧 ∈ ran (ball‘𝐷)(𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝐴) ↔ ∃𝑦 ∈ ℝ+ (𝑥(ball‘𝐷)𝑦) ⊆ 𝐴)) | |
5 | 3, 4 | sylan2 586 | . . . . 5 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝑥 ∈ 𝐴)) → (∃𝑧 ∈ ran (ball‘𝐷)(𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝐴) ↔ ∃𝑦 ∈ ℝ+ (𝑥(ball‘𝐷)𝑦) ⊆ 𝐴)) |
6 | 5 | anassrs 461 | . . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑥 ∈ 𝐴) → (∃𝑧 ∈ ran (ball‘𝐷)(𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝐴) ↔ ∃𝑦 ∈ ℝ+ (𝑥(ball‘𝐷)𝑦) ⊆ 𝐴)) |
7 | 6 | ralbidva 3194 | . . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (∀𝑥 ∈ 𝐴 ∃𝑧 ∈ ran (ball‘𝐷)(𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝐴) ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ ℝ+ (𝑥(ball‘𝐷)𝑦) ⊆ 𝐴)) |
8 | 7 | pm5.32da 574 | . 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → ((𝐴 ⊆ 𝑋 ∧ ∀𝑥 ∈ 𝐴 ∃𝑧 ∈ ran (ball‘𝐷)(𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝐴)) ↔ (𝐴 ⊆ 𝑋 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ ℝ+ (𝑥(ball‘𝐷)𝑦) ⊆ 𝐴))) |
9 | 2, 8 | bitrd 271 | 1 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐴 ∈ 𝐽 ↔ (𝐴 ⊆ 𝑋 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ ℝ+ (𝑥(ball‘𝐷)𝑦) ⊆ 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1656 ∈ wcel 2164 ∀wral 3117 ∃wrex 3118 ⊆ wss 3798 ran crn 5347 ‘cfv 6127 (class class class)co 6910 ℝ+crp 12119 ∞Metcxmet 20098 ballcbl 20100 MetOpencmopn 20103 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 ax-pre-sup 10337 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-om 7332 df-1st 7433 df-2nd 7434 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-er 8014 df-map 8129 df-en 8229 df-dom 8230 df-sdom 8231 df-sup 8623 df-inf 8624 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-div 11017 df-nn 11358 df-2 11421 df-n0 11626 df-z 11712 df-uz 11976 df-q 12079 df-rp 12120 df-xneg 12239 df-xadd 12240 df-xmul 12241 df-topgen 16464 df-psmet 20105 df-xmet 20106 df-bl 20108 df-mopn 20109 df-bases 21128 |
This theorem is referenced by: metrest 22706 tgioo 22976 xrsmopn 22992 recld2 22994 tpr2rico 30499 dya2icoseg2 30881 opnrebl 32848 opnrebl2 32849 |
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