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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 | β’ π½ = (MetOpenβπ·) |
Ref | Expression |
---|---|
mopnuni | β’ (π· β (βMetβπ) β π = βͺ π½) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mopnval.1 | . . 3 β’ π½ = (MetOpenβπ·) | |
2 | 1 | mopntopon 23808 | . 2 β’ (π· β (βMetβπ) β π½ β (TopOnβπ)) |
3 | toponuni 22279 | . 2 β’ (π½ β (TopOnβπ) β π = βͺ π½) | |
4 | 2, 3 | syl 17 | 1 β’ (π· β (βMetβπ) β π = βͺ π½) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 βͺ cuni 4870 βcfv 6501 βMetcxmet 20797 MetOpencmopn 20802 TopOnctopon 22275 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 ax-pre-sup 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-er 8655 df-map 8774 df-en 8891 df-dom 8892 df-sdom 8893 df-sup 9385 df-inf 9386 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-div 11820 df-nn 12161 df-2 12223 df-n0 12421 df-z 12507 df-uz 12771 df-q 12881 df-rp 12923 df-xneg 13040 df-xadd 13041 df-xmul 13042 df-topgen 17332 df-psmet 20804 df-xmet 20805 df-bl 20807 df-mopn 20808 df-top 22259 df-topon 22276 df-bases 22312 |
This theorem is referenced by: mopnfss 23812 setsmstopn 23849 neibl 23873 lpbl 23875 blcld 23877 met1stc 23893 met2ndci 23894 met2ndc 23895 metcnpi 23916 metcnpi2 23917 metcnpi3 23918 tngtopn 24030 recld2 24193 xmetdcn 24217 metnrmlem1a 24237 metnrmlem1 24238 metnrmlem2 24239 metnrmlem3 24240 lebnumlem1 24340 lebnumlem3 24342 lebnum 24343 metelcls 24685 metcld 24686 flimcfil 24694 metsscmetcld 24695 cmetss 24696 cmpcmet 24699 bcthlem2 24705 bcthlem4 24707 bcthlem5 24708 bcth3 24711 heicant 36142 heibor1lem 36297 heibor1 36298 heiborlem3 36301 heiborlem8 36306 heiborlem10 36308 heibor 36309 |
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