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Theorem dscopn 20962
Description: The discrete metric generates the discrete topology. In particular, the discrete topology is metrizable. (Contributed by Mario Carneiro, 29-Jan-2014.)
Hypothesis
Ref Expression
dscmet.1  |-  D  =  ( x  e.  X ,  y  e.  X  |->  if ( x  =  y ,  0 ,  1 ) )
Assertion
Ref Expression
dscopn  |-  ( X  e.  V  ->  ( MetOpen
`  D )  =  ~P X )
Distinct variable group:    x, y, X
Allowed substitution hints:    D( x, y)    V( x, y)

Proof of Theorem dscopn
Dummy variables  v  u  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dscmet.1 . . . . . . 7  |-  D  =  ( x  e.  X ,  y  e.  X  |->  if ( x  =  y ,  0 ,  1 ) )
21dscmet 20961 . . . . . 6  |-  ( X  e.  V  ->  D  e.  ( Met `  X
) )
3 metxmet 20705 . . . . . 6  |-  ( D  e.  ( Met `  X
)  ->  D  e.  ( *Met `  X
) )
42, 3syl 16 . . . . 5  |-  ( X  e.  V  ->  D  e.  ( *Met `  X ) )
5 eqid 2467 . . . . . 6  |-  ( MetOpen `  D )  =  (
MetOpen `  D )
65elmopn 20813 . . . . 5  |-  ( D  e.  ( *Met `  X )  ->  (
u  e.  ( MetOpen `  D )  <->  ( u  C_  X  /\  A. v  e.  u  E. w  e.  ran  ( ball `  D
) ( v  e.  w  /\  w  C_  u ) ) ) )
74, 6syl 16 . . . 4  |-  ( X  e.  V  ->  (
u  e.  ( MetOpen `  D )  <->  ( u  C_  X  /\  A. v  e.  u  E. w  e.  ran  ( ball `  D
) ( v  e.  w  /\  w  C_  u ) ) ) )
8 simpll 753 . . . . . . . . 9  |-  ( ( ( X  e.  V  /\  u  C_  X )  /\  v  e.  u
)  ->  X  e.  V )
9 ssel2 3504 . . . . . . . . . 10  |-  ( ( u  C_  X  /\  v  e.  u )  ->  v  e.  X )
109adantll 713 . . . . . . . . 9  |-  ( ( ( X  e.  V  /\  u  C_  X )  /\  v  e.  u
)  ->  v  e.  X )
118, 10jca 532 . . . . . . . 8  |-  ( ( ( X  e.  V  /\  u  C_  X )  /\  v  e.  u
)  ->  ( X  e.  V  /\  v  e.  X ) )
12 elsn 4047 . . . . . . . . . . . 12  |-  ( w  e.  { v }  <-> 
w  =  v )
13 eleq1a 2550 . . . . . . . . . . . . . . 15  |-  ( v  e.  X  ->  (
w  =  v  ->  w  e.  X )
)
14 simpl 457 . . . . . . . . . . . . . . . 16  |-  ( ( w  e.  X  /\  ( v D w )  <  1 )  ->  w  e.  X
)
1514a1i 11 . . . . . . . . . . . . . . 15  |-  ( v  e.  X  ->  (
( w  e.  X  /\  ( v D w )  <  1 )  ->  w  e.  X
) )
16 eqeq12 2486 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( x  =  v  /\  y  =  w )  ->  ( x  =  y  <-> 
v  =  w ) )
1716ifbid 3967 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( x  =  v  /\  y  =  w )  ->  if ( x  =  y ,  0 ,  1 )  =  if ( v  =  w ,  0 ,  1 ) )
18 0re 9608 . . . . . . . . . . . . . . . . . . . . . 22  |-  0  e.  RR
19 1re 9607 . . . . . . . . . . . . . . . . . . . . . 22  |-  1  e.  RR
2018, 19keepel 4013 . . . . . . . . . . . . . . . . . . . . 21  |-  if ( v  =  w ,  0 ,  1 )  e.  RR
2120elexi 3128 . . . . . . . . . . . . . . . . . . . 20  |-  if ( v  =  w ,  0 ,  1 )  e.  _V
2217, 1, 21ovmpt2a 6428 . . . . . . . . . . . . . . . . . . 19  |-  ( ( v  e.  X  /\  w  e.  X )  ->  ( v D w )  =  if ( v  =  w ,  0 ,  1 ) )
2322breq1d 4463 . . . . . . . . . . . . . . . . . 18  |-  ( ( v  e.  X  /\  w  e.  X )  ->  ( ( v D w )  <  1  <->  if ( v  =  w ,  0 ,  1 )  <  1 ) )
2419ltnri 9705 . . . . . . . . . . . . . . . . . . . . . 22  |-  -.  1  <  1
25 iffalse 3954 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( -.  v  =  w  ->  if ( v  =  w ,  0 ,  1 )  =  1 )
2625breq1d 4463 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( -.  v  =  w  -> 
( if ( v  =  w ,  0 ,  1 )  <  1  <->  1  <  1
) )
2724, 26mtbiri 303 . . . . . . . . . . . . . . . . . . . . 21  |-  ( -.  v  =  w  ->  -.  if ( v  =  w ,  0 ,  1 )  <  1
)
2827con4i 130 . . . . . . . . . . . . . . . . . . . 20  |-  ( if ( v  =  w ,  0 ,  1 )  <  1  -> 
v  =  w )
29 iftrue 3951 . . . . . . . . . . . . . . . . . . . . 21  |-  ( v  =  w  ->  if ( v  =  w ,  0 ,  1 )  =  0 )
30 0lt1 10087 . . . . . . . . . . . . . . . . . . . . 21  |-  0  <  1
3129, 30syl6eqbr 4490 . . . . . . . . . . . . . . . . . . . 20  |-  ( v  =  w  ->  if ( v  =  w ,  0 ,  1 )  <  1 )
3228, 31impbii 188 . . . . . . . . . . . . . . . . . . 19  |-  ( if ( v  =  w ,  0 ,  1 )  <  1  <->  v  =  w )
33 equcom 1743 . . . . . . . . . . . . . . . . . . 19  |-  ( v  =  w  <->  w  =  v )
3432, 33bitri 249 . . . . . . . . . . . . . . . . . 18  |-  ( if ( v  =  w ,  0 ,  1 )  <  1  <->  w  =  v )
3523, 34syl6rbb 262 . . . . . . . . . . . . . . . . 17  |-  ( ( v  e.  X  /\  w  e.  X )  ->  ( w  =  v  <-> 
( v D w )  <  1 ) )
36 simpr 461 . . . . . . . . . . . . . . . . . 18  |-  ( ( v  e.  X  /\  w  e.  X )  ->  w  e.  X )
3736biantrurd 508 . . . . . . . . . . . . . . . . 17  |-  ( ( v  e.  X  /\  w  e.  X )  ->  ( ( v D w )  <  1  <->  ( w  e.  X  /\  ( v D w )  <  1 ) ) )
3835, 37bitrd 253 . . . . . . . . . . . . . . . 16  |-  ( ( v  e.  X  /\  w  e.  X )  ->  ( w  =  v  <-> 
( w  e.  X  /\  ( v D w )  <  1 ) ) )
3938ex 434 . . . . . . . . . . . . . . 15  |-  ( v  e.  X  ->  (
w  e.  X  -> 
( w  =  v  <-> 
( w  e.  X  /\  ( v D w )  <  1 ) ) ) )
4013, 15, 39pm5.21ndd 354 . . . . . . . . . . . . . 14  |-  ( v  e.  X  ->  (
w  =  v  <->  ( w  e.  X  /\  (
v D w )  <  1 ) ) )
4140adantl 466 . . . . . . . . . . . . 13  |-  ( ( X  e.  V  /\  v  e.  X )  ->  ( w  =  v  <-> 
( w  e.  X  /\  ( v D w )  <  1 ) ) )
42 1rp 11236 . . . . . . . . . . . . . . . 16  |-  1  e.  RR+
43 rpxr 11239 . . . . . . . . . . . . . . . 16  |-  ( 1  e.  RR+  ->  1  e. 
RR* )
4442, 43ax-mp 5 . . . . . . . . . . . . . . 15  |-  1  e.  RR*
45 elbl 20759 . . . . . . . . . . . . . . 15  |-  ( ( D  e.  ( *Met `  X )  /\  v  e.  X  /\  1  e.  RR* )  ->  ( w  e.  ( v ( ball `  D
) 1 )  <->  ( w  e.  X  /\  (
v D w )  <  1 ) ) )
4644, 45mp3an3 1313 . . . . . . . . . . . . . 14  |-  ( ( D  e.  ( *Met `  X )  /\  v  e.  X
)  ->  ( w  e.  ( v ( ball `  D ) 1 )  <-> 
( w  e.  X  /\  ( v D w )  <  1 ) ) )
474, 46sylan 471 . . . . . . . . . . . . 13  |-  ( ( X  e.  V  /\  v  e.  X )  ->  ( w  e.  ( v ( ball `  D
) 1 )  <->  ( w  e.  X  /\  (
v D w )  <  1 ) ) )
4841, 47bitr4d 256 . . . . . . . . . . . 12  |-  ( ( X  e.  V  /\  v  e.  X )  ->  ( w  =  v  <-> 
w  e.  ( v ( ball `  D
) 1 ) ) )
4912, 48syl5bb 257 . . . . . . . . . . 11  |-  ( ( X  e.  V  /\  v  e.  X )  ->  ( w  e.  {
v }  <->  w  e.  ( v ( ball `  D ) 1 ) ) )
5049eqrdv 2464 . . . . . . . . . 10  |-  ( ( X  e.  V  /\  v  e.  X )  ->  { v }  =  ( v ( ball `  D ) 1 ) )
51 blelrn 20788 . . . . . . . . . . . 12  |-  ( ( D  e.  ( *Met `  X )  /\  v  e.  X  /\  1  e.  RR* )  ->  ( v ( ball `  D ) 1 )  e.  ran  ( ball `  D ) )
5244, 51mp3an3 1313 . . . . . . . . . . 11  |-  ( ( D  e.  ( *Met `  X )  /\  v  e.  X
)  ->  ( v
( ball `  D )
1 )  e.  ran  ( ball `  D )
)
534, 52sylan 471 . . . . . . . . . 10  |-  ( ( X  e.  V  /\  v  e.  X )  ->  ( v ( ball `  D ) 1 )  e.  ran  ( ball `  D ) )
5450, 53eqeltrd 2555 . . . . . . . . 9  |-  ( ( X  e.  V  /\  v  e.  X )  ->  { v }  e.  ran  ( ball `  D
) )
55 snssi 4177 . . . . . . . . . 10  |-  ( v  e.  u  ->  { v }  C_  u )
56 ssnid 4062 . . . . . . . . . 10  |-  v  e. 
{ v }
5755, 56jctil 537 . . . . . . . . 9  |-  ( v  e.  u  ->  (
v  e.  { v }  /\  { v }  C_  u )
)
58 eleq2 2540 . . . . . . . . . . 11  |-  ( w  =  { v }  ->  ( v  e.  w  <->  v  e.  {
v } ) )
59 sseq1 3530 . . . . . . . . . . 11  |-  ( w  =  { v }  ->  ( w  C_  u 
<->  { v }  C_  u ) )
6058, 59anbi12d 710 . . . . . . . . . 10  |-  ( w  =  { v }  ->  ( ( v  e.  w  /\  w  C_  u )  <->  ( v  e.  { v }  /\  { v }  C_  u
) ) )
6160rspcev 3219 . . . . . . . . 9  |-  ( ( { v }  e.  ran  ( ball `  D
)  /\  ( v  e.  { v }  /\  { v }  C_  u
) )  ->  E. w  e.  ran  ( ball `  D
) ( v  e.  w  /\  w  C_  u ) )
6254, 57, 61syl2an 477 . . . . . . . 8  |-  ( ( ( X  e.  V  /\  v  e.  X
)  /\  v  e.  u )  ->  E. w  e.  ran  ( ball `  D
) ( v  e.  w  /\  w  C_  u ) )
6311, 62sylancom 667 . . . . . . 7  |-  ( ( ( X  e.  V  /\  u  C_  X )  /\  v  e.  u
)  ->  E. w  e.  ran  ( ball `  D
) ( v  e.  w  /\  w  C_  u ) )
6463ralrimiva 2881 . . . . . 6  |-  ( ( X  e.  V  /\  u  C_  X )  ->  A. v  e.  u  E. w  e.  ran  ( ball `  D )
( v  e.  w  /\  w  C_  u ) )
6564ex 434 . . . . 5  |-  ( X  e.  V  ->  (
u  C_  X  ->  A. v  e.  u  E. w  e.  ran  ( ball `  D ) ( v  e.  w  /\  w  C_  u ) ) )
6665pm4.71d 634 . . . 4  |-  ( X  e.  V  ->  (
u  C_  X  <->  ( u  C_  X  /\  A. v  e.  u  E. w  e.  ran  ( ball `  D
) ( v  e.  w  /\  w  C_  u ) ) ) )
677, 66bitr4d 256 . . 3  |-  ( X  e.  V  ->  (
u  e.  ( MetOpen `  D )  <->  u  C_  X
) )
68 selpw 4023 . . 3  |-  ( u  e.  ~P X  <->  u  C_  X
)
6967, 68syl6bbr 263 . 2  |-  ( X  e.  V  ->  (
u  e.  ( MetOpen `  D )  <->  u  e.  ~P X ) )
7069eqrdv 2464 1  |-  ( X  e.  V  ->  ( MetOpen
`  D )  =  ~P X )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2817   E.wrex 2818    C_ wss 3481   ifcif 3945   ~Pcpw 4016   {csn 4033   class class class wbr 4453   ran crn 5006   ` cfv 5594  (class class class)co 6295    |-> cmpt2 6297   RRcr 9503   0cc0 9504   1c1 9505   RR*cxr 9639    < clt 9640   RR+crp 11232   *Metcxmt 18273   Metcme 18274   ballcbl 18275   MetOpencmopn 18278
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-er 7323  df-map 7434  df-en 7529  df-dom 7530  df-sdom 7531  df-sup 7913  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-n0 10808  df-z 10877  df-uz 11095  df-q 11195  df-rp 11233  df-xneg 11330  df-xadd 11331  df-xmul 11332  df-topgen 14716  df-psmet 18281  df-xmet 18282  df-met 18283  df-bl 18284  df-mopn 18285  df-bases 19270
This theorem is referenced by: (None)
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