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Mirrors > Home > ILE Home > Th. List > mopnval | GIF version |
Description: An open set is a subset of a metric space which includes a ball around each of its points. Definition 1.3-2 of [Kreyszig] p. 18. The object (MetOpen‘𝐷) is the family of all open sets in the metric space determined by the metric 𝐷. By mopntop 12616, the open sets of a metric space form a topology 𝐽, whose base set is ∪ 𝐽 by mopnuni 12617. (Contributed by NM, 1-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.) |
Ref | Expression |
---|---|
mopnval.1 | ⊢ 𝐽 = (MetOpen‘𝐷) |
Ref | Expression |
---|---|
mopnval | ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 = (topGen‘ran (ball‘𝐷))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mopnval.1 | . 2 ⊢ 𝐽 = (MetOpen‘𝐷) | |
2 | df-mopn 12163 | . . 3 ⊢ MetOpen = (𝑑 ∈ ∪ ran ∞Met ↦ (topGen‘ran (ball‘𝑑))) | |
3 | fveq2 5421 | . . . . 5 ⊢ (𝑑 = 𝐷 → (ball‘𝑑) = (ball‘𝐷)) | |
4 | 3 | rneqd 4768 | . . . 4 ⊢ (𝑑 = 𝐷 → ran (ball‘𝑑) = ran (ball‘𝐷)) |
5 | 4 | fveq2d 5425 | . . 3 ⊢ (𝑑 = 𝐷 → (topGen‘ran (ball‘𝑑)) = (topGen‘ran (ball‘𝐷))) |
6 | xmetrel 12515 | . . . . . . . 8 ⊢ Rel ∞Met | |
7 | relelfvdm 5453 | . . . . . . . 8 ⊢ ((Rel ∞Met ∧ 𝐷 ∈ (∞Met‘𝑋)) → 𝑋 ∈ dom ∞Met) | |
8 | 6, 7 | mpan 420 | . . . . . . 7 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 ∈ dom ∞Met) |
9 | 8 | elexd 2699 | . . . . . 6 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 ∈ V) |
10 | fvssunirng 5436 | . . . . . 6 ⊢ (𝑋 ∈ V → (∞Met‘𝑋) ⊆ ∪ ran ∞Met) | |
11 | 9, 10 | syl 14 | . . . . 5 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (∞Met‘𝑋) ⊆ ∪ ran ∞Met) |
12 | 11 | sseld 3096 | . . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐷 ∈ (∞Met‘𝑋) → 𝐷 ∈ ∪ ran ∞Met)) |
13 | 12 | pm2.43i 49 | . . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷 ∈ ∪ ran ∞Met) |
14 | blbas 12605 | . . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → ran (ball‘𝐷) ∈ TopBases) | |
15 | tgcl 12236 | . . . 4 ⊢ (ran (ball‘𝐷) ∈ TopBases → (topGen‘ran (ball‘𝐷)) ∈ Top) | |
16 | 14, 15 | syl 14 | . . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (topGen‘ran (ball‘𝐷)) ∈ Top) |
17 | 2, 5, 13, 16 | fvmptd3 5514 | . 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (MetOpen‘𝐷) = (topGen‘ran (ball‘𝐷))) |
18 | 1, 17 | syl5eq 2184 | 1 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 = (topGen‘ran (ball‘𝐷))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ∈ wcel 1480 Vcvv 2686 ⊆ wss 3071 ∪ cuni 3736 dom cdm 4539 ran crn 4540 Rel wrel 4544 ‘cfv 5123 topGenctg 12138 ∞Metcxmet 12152 ballcbl 12154 MetOpencmopn 12157 Topctop 12167 TopBasesctb 12212 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 ax-cnex 7714 ax-resscn 7715 ax-1cn 7716 ax-1re 7717 ax-icn 7718 ax-addcl 7719 ax-addrcl 7720 ax-mulcl 7721 ax-mulrcl 7722 ax-addcom 7723 ax-mulcom 7724 ax-addass 7725 ax-mulass 7726 ax-distr 7727 ax-i2m1 7728 ax-0lt1 7729 ax-1rid 7730 ax-0id 7731 ax-rnegex 7732 ax-precex 7733 ax-cnre 7734 ax-pre-ltirr 7735 ax-pre-ltwlin 7736 ax-pre-lttrn 7737 ax-pre-apti 7738 ax-pre-ltadd 7739 ax-pre-mulgt0 7740 ax-pre-mulext 7741 ax-arch 7742 ax-caucvg 7743 |
This theorem depends on definitions: df-bi 116 df-stab 816 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-po 4218 df-iso 4219 df-iord 4288 df-on 4290 df-ilim 4291 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-isom 5132 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-frec 6288 df-map 6544 df-sup 6871 df-inf 6872 df-pnf 7805 df-mnf 7806 df-xr 7807 df-ltxr 7808 df-le 7809 df-sub 7938 df-neg 7939 df-reap 8340 df-ap 8347 df-div 8436 df-inn 8724 df-2 8782 df-3 8783 df-4 8784 df-n0 8981 df-z 9058 df-uz 9330 df-q 9415 df-rp 9445 df-xneg 9562 df-xadd 9563 df-seqfrec 10222 df-exp 10296 df-cj 10617 df-re 10618 df-im 10619 df-rsqrt 10773 df-abs 10774 df-topgen 12144 df-psmet 12159 df-xmet 12160 df-bl 12162 df-mopn 12163 df-top 12168 df-bases 12213 |
This theorem is referenced by: mopntopon 12615 elmopn 12618 blssopn 12657 metss 12666 xmettxlem 12681 xmettx 12682 metcnp3 12683 tgioo 12718 |
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