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Theorem setsmstopn 18024
Description: The topology of a constructed metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
Hypotheses
Ref Expression
setsms.x  |-  ( ph  ->  X  =  ( Base `  M ) )
setsms.d  |-  ( ph  ->  D  =  ( (
dist `  M )  |`  ( X  X.  X
) ) )
setsms.k  |-  ( ph  ->  K  =  ( M sSet  <. (TopSet `  ndx ) ,  ( MetOpen `  D ) >. ) )
setsms.m  |-  ( ph  ->  M  e.  V )
Assertion
Ref Expression
setsmstopn  |-  ( ph  ->  ( MetOpen `  D )  =  ( TopOpen `  K
) )

Proof of Theorem setsmstopn
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 setsms.x . . 3  |-  ( ph  ->  X  =  ( Base `  M ) )
2 setsms.d . . 3  |-  ( ph  ->  D  =  ( (
dist `  M )  |`  ( X  X.  X
) ) )
3 setsms.k . . 3  |-  ( ph  ->  K  =  ( M sSet  <. (TopSet `  ndx ) ,  ( MetOpen `  D ) >. ) )
4 setsms.m . . 3  |-  ( ph  ->  M  e.  V )
51, 2, 3, 4setsmstset 18023 . 2  |-  ( ph  ->  ( MetOpen `  D )  =  (TopSet `  K )
)
6 df-mopn 16376 . . . . . . . 8  |-  MetOpen  =  ( x  e.  U. ran  * Met  |->  ( topGen `  ran  ( ball `  x )
) )
76dmmptss 5169 . . . . . . 7  |-  dom  MetOpen  C_  U. ran  * Met
87sseli 3176 . . . . . 6  |-  ( D  e.  dom  MetOpen  ->  D  e.  U. ran  * Met )
9 simpr 447 . . . . . . . . . . 11  |-  ( (
ph  /\  D  e.  U.
ran  * Met )  ->  D  e.  U. ran  * Met )
10 xmetunirn 17902 . . . . . . . . . . 11  |-  ( D  e.  U. ran  * Met 
<->  D  e.  ( * Met `  dom  dom  D ) )
119, 10sylib 188 . . . . . . . . . 10  |-  ( (
ph  /\  D  e.  U.
ran  * Met )  ->  D  e.  ( * Met `  dom  dom  D
) )
12 eqid 2283 . . . . . . . . . . 11  |-  ( MetOpen `  D )  =  (
MetOpen `  D )
1312mopnuni 17987 . . . . . . . . . 10  |-  ( D  e.  ( * Met ` 
dom  dom  D )  ->  dom  dom  D  =  U. ( MetOpen `  D )
)
1411, 13syl 15 . . . . . . . . 9  |-  ( (
ph  /\  D  e.  U.
ran  * Met )  ->  dom  dom  D  =  U. ( MetOpen `  D )
)
15 inss1 3389 . . . . . . . . . . . . 13  |-  ( ( X  X.  X )  i^i  dom  ( dist `  M ) )  C_  ( X  X.  X
)
162dmeqd 4881 . . . . . . . . . . . . . . 15  |-  ( ph  ->  dom  D  =  dom  ( ( dist `  M
)  |`  ( X  X.  X ) ) )
17 dmres 4976 . . . . . . . . . . . . . . 15  |-  dom  (
( dist `  M )  |`  ( X  X.  X
) )  =  ( ( X  X.  X
)  i^i  dom  ( dist `  M ) )
1816, 17syl6eq 2331 . . . . . . . . . . . . . 14  |-  ( ph  ->  dom  D  =  ( ( X  X.  X
)  i^i  dom  ( dist `  M ) ) )
1918sseq1d 3205 . . . . . . . . . . . . 13  |-  ( ph  ->  ( dom  D  C_  ( X  X.  X
)  <->  ( ( X  X.  X )  i^i 
dom  ( dist `  M
) )  C_  ( X  X.  X ) ) )
2015, 19mpbiri 224 . . . . . . . . . . . 12  |-  ( ph  ->  dom  D  C_  ( X  X.  X ) )
21 dmss 4878 . . . . . . . . . . . 12  |-  ( dom 
D  C_  ( X  X.  X )  ->  dom  dom 
D  C_  dom  ( X  X.  X ) )
2220, 21syl 15 . . . . . . . . . . 11  |-  ( ph  ->  dom  dom  D  C_  dom  ( X  X.  X
) )
23 dmxpid 4898 . . . . . . . . . . 11  |-  dom  ( X  X.  X )  =  X
2422, 23syl6sseq 3224 . . . . . . . . . 10  |-  ( ph  ->  dom  dom  D  C_  X
)
2524adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  D  e.  U.
ran  * Met )  ->  dom  dom  D  C_  X
)
2614, 25eqsstr3d 3213 . . . . . . . 8  |-  ( (
ph  /\  D  e.  U.
ran  * Met )  ->  U. ( MetOpen `  D )  C_  X )
27 sspwuni 3987 . . . . . . . 8  |-  ( (
MetOpen `  D )  C_  ~P X  <->  U. ( MetOpen `  D
)  C_  X )
2826, 27sylibr 203 . . . . . . 7  |-  ( (
ph  /\  D  e.  U.
ran  * Met )  -> 
( MetOpen `  D )  C_ 
~P X )
2928ex 423 . . . . . 6  |-  ( ph  ->  ( D  e.  U. ran  * Met  ->  ( MetOpen
`  D )  C_  ~P X ) )
308, 29syl5 28 . . . . 5  |-  ( ph  ->  ( D  e.  dom  MetOpen  ->  ( MetOpen `  D )  C_ 
~P X ) )
31 0ss 3483 . . . . . 6  |-  (/)  C_  ~P X
32 ndmfv 5552 . . . . . . 7  |-  ( -.  D  e.  dom  MetOpen  ->  ( MetOpen
`  D )  =  (/) )
3332sseq1d 3205 . . . . . 6  |-  ( -.  D  e.  dom  MetOpen  ->  (
( MetOpen `  D )  C_ 
~P X  <->  (/)  C_  ~P X ) )
3431, 33mpbiri 224 . . . . 5  |-  ( -.  D  e.  dom  MetOpen  ->  ( MetOpen
`  D )  C_  ~P X )
3530, 34pm2.61d1 151 . . . 4  |-  ( ph  ->  ( MetOpen `  D )  C_ 
~P X )
361, 2, 3setsmsbas 18021 . . . . 5  |-  ( ph  ->  X  =  ( Base `  K ) )
3736pweqd 3630 . . . 4  |-  ( ph  ->  ~P X  =  ~P ( Base `  K )
)
3835, 5, 373sstr3d 3220 . . 3  |-  ( ph  ->  (TopSet `  K )  C_ 
~P ( Base `  K
) )
39 eqid 2283 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
40 eqid 2283 . . . 4  |-  (TopSet `  K )  =  (TopSet `  K )
4139, 40topnid 13340 . . 3  |-  ( (TopSet `  K )  C_  ~P ( Base `  K )  ->  (TopSet `  K )  =  ( TopOpen `  K
) )
4238, 41syl 15 . 2  |-  ( ph  ->  (TopSet `  K )  =  ( TopOpen `  K
) )
435, 42eqtrd 2315 1  |-  ( ph  ->  ( MetOpen `  D )  =  ( TopOpen `  K
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    i^i cin 3151    C_ wss 3152   (/)c0 3455   ~Pcpw 3625   <.cop 3643   U.cuni 3827    X. cxp 4687   dom cdm 4689   ran crn 4690    |` cres 4691   ` cfv 5255  (class class class)co 5858   ndxcnx 13145   sSet csts 13146   Basecbs 13148  TopSetcts 13214   distcds 13217   TopOpenctopn 13326   topGenctg 13342   * Metcxmt 16369   ballcbl 16371   MetOpencmopn 16372
This theorem is referenced by:  setsxms  18025  tmslem  18028
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-tset 13227  df-rest 13327  df-topn 13328  df-topgen 13344  df-xmet 16373  df-bl 16375  df-mopn 16376  df-top 16636  df-bases 16638  df-topon 16639
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