Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > mopni | 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, 3-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.) |
Ref | Expression |
---|---|
mopni.1 | ⊢ 𝐽 = (MetOpen‘𝐷) |
Ref | Expression |
---|---|
mopni | ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝐽 ∧ 𝑃 ∈ 𝐴) → ∃𝑥 ∈ ran (ball‘𝐷)(𝑃 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mopni.1 | . . . . 5 ⊢ 𝐽 = (MetOpen‘𝐷) | |
2 | 1 | elmopn 22247 | . . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐴 ∈ 𝐽 ↔ (𝐴 ⊆ 𝑋 ∧ ∀𝑦 ∈ 𝐴 ∃𝑥 ∈ ran (ball‘𝐷)(𝑦 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐴)))) |
3 | 2 | simplbda 654 | . . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝐽) → ∀𝑦 ∈ 𝐴 ∃𝑥 ∈ ran (ball‘𝐷)(𝑦 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐴)) |
4 | eleq1 2689 | . . . . . 6 ⊢ (𝑦 = 𝑃 → (𝑦 ∈ 𝑥 ↔ 𝑃 ∈ 𝑥)) | |
5 | 4 | anbi1d 741 | . . . . 5 ⊢ (𝑦 = 𝑃 → ((𝑦 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐴) ↔ (𝑃 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐴))) |
6 | 5 | rexbidv 3052 | . . . 4 ⊢ (𝑦 = 𝑃 → (∃𝑥 ∈ ran (ball‘𝐷)(𝑦 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐴) ↔ ∃𝑥 ∈ ran (ball‘𝐷)(𝑃 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐴))) |
7 | 6 | rspccv 3306 | . . 3 ⊢ (∀𝑦 ∈ 𝐴 ∃𝑥 ∈ ran (ball‘𝐷)(𝑦 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐴) → (𝑃 ∈ 𝐴 → ∃𝑥 ∈ ran (ball‘𝐷)(𝑃 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐴))) |
8 | 3, 7 | syl 17 | . 2 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝐽) → (𝑃 ∈ 𝐴 → ∃𝑥 ∈ ran (ball‘𝐷)(𝑃 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐴))) |
9 | 8 | 3impia 1261 | 1 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝐽 ∧ 𝑃 ∈ 𝐴) → ∃𝑥 ∈ ran (ball‘𝐷)(𝑃 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ∃wrex 2913 ⊆ wss 3574 ran crn 5115 ‘cfv 5888 ∞Metcxmt 19731 ballcbl 19733 MetOpencmopn 19736 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-inf 8349 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-n0 11293 df-z 11378 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-topgen 16104 df-psmet 19738 df-xmet 19739 df-bl 19741 df-mopn 19742 df-bases 20750 |
This theorem is referenced by: mopni2 22298 |
Copyright terms: Public domain | W3C validator |