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Mirrors > Home > MPE Home > Th. List > mopnuni | Structured version Visualization version GIF version |
Description: The union of all open sets in a metric space is its underlying set. (Contributed by NM, 4-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.) |
Ref | Expression |
---|---|
mopnval.1 | ⊢ 𝐽 = (MetOpen‘𝐷) |
Ref | Expression |
---|---|
mopnuni | ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = ∪ 𝐽) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mopnval.1 | . . 3 ⊢ 𝐽 = (MetOpen‘𝐷) | |
2 | 1 | mopntopon 23185 | . 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ (TopOn‘𝑋)) |
3 | toponuni 21658 | . 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽) | |
4 | 2, 3 | syl 17 | 1 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = ∪ 𝐽) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2113 ∪ cuni 4793 ‘cfv 6333 ∞Metcxmet 20195 MetOpencmopn 20200 TopOnctopon 21654 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 ax-cnex 10664 ax-resscn 10665 ax-1cn 10666 ax-icn 10667 ax-addcl 10668 ax-addrcl 10669 ax-mulcl 10670 ax-mulrcl 10671 ax-mulcom 10672 ax-addass 10673 ax-mulass 10674 ax-distr 10675 ax-i2m1 10676 ax-1ne0 10677 ax-1rid 10678 ax-rnegex 10679 ax-rrecex 10680 ax-cnre 10681 ax-pre-lttri 10682 ax-pre-lttrn 10683 ax-pre-ltadd 10684 ax-pre-mulgt0 10685 ax-pre-sup 10686 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-pss 3860 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-tp 4518 df-op 4520 df-uni 4794 df-iun 4880 df-br 5028 df-opab 5090 df-mpt 5108 df-tr 5134 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6123 df-ord 6169 df-on 6170 df-lim 6171 df-suc 6172 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-riota 7121 df-ov 7167 df-oprab 7168 df-mpo 7169 df-om 7594 df-1st 7707 df-2nd 7708 df-wrecs 7969 df-recs 8030 df-rdg 8068 df-er 8313 df-map 8432 df-en 8549 df-dom 8550 df-sdom 8551 df-sup 8972 df-inf 8973 df-pnf 10748 df-mnf 10749 df-xr 10750 df-ltxr 10751 df-le 10752 df-sub 10943 df-neg 10944 df-div 11369 df-nn 11710 df-2 11772 df-n0 11970 df-z 12056 df-uz 12318 df-q 12424 df-rp 12466 df-xneg 12583 df-xadd 12584 df-xmul 12585 df-topgen 16813 df-psmet 20202 df-xmet 20203 df-bl 20205 df-mopn 20206 df-top 21638 df-topon 21655 df-bases 21690 |
This theorem is referenced by: mopnfss 23189 setsmstopn 23224 neibl 23247 lpbl 23249 blcld 23251 met1stc 23267 met2ndci 23268 met2ndc 23269 metcnpi 23290 metcnpi2 23291 metcnpi3 23292 tngtopn 23396 recld2 23559 xmetdcn 23583 metnrmlem1a 23603 metnrmlem1 23604 metnrmlem2 23605 metnrmlem3 23606 lebnumlem1 23706 lebnumlem3 23708 lebnum 23709 metelcls 24050 metcld 24051 flimcfil 24059 metsscmetcld 24060 cmetss 24061 cmpcmet 24064 bcthlem2 24070 bcthlem4 24072 bcthlem5 24073 bcth3 24076 heicant 35424 heibor1lem 35579 heibor1 35580 heiborlem3 35583 heiborlem8 35588 heiborlem10 35590 heibor 35591 |
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