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Theorem dscopn 21666
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 21665 . . . . . 6  |-  ( X  e.  V  ->  D  e.  ( Met `  X
) )
3 metxmet 21427 . . . . . 6  |-  ( D  e.  ( Met `  X
)  ->  D  e.  ( *Met `  X
) )
42, 3syl 17 . . . . 5  |-  ( X  e.  V  ->  D  e.  ( *Met `  X ) )
5 eqid 2471 . . . . . 6  |-  ( MetOpen `  D )  =  (
MetOpen `  D )
65elmopn 21535 . . . . 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 17 . . . 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 768 . . . . . . . . 9  |-  ( ( ( X  e.  V  /\  u  C_  X )  /\  v  e.  u
)  ->  X  e.  V )
9 ssel2 3413 . . . . . . . . . 10  |-  ( ( u  C_  X  /\  v  e.  u )  ->  v  e.  X )
109adantll 728 . . . . . . . . 9  |-  ( ( ( X  e.  V  /\  u  C_  X )  /\  v  e.  u
)  ->  v  e.  X )
118, 10jca 541 . . . . . . . 8  |-  ( ( ( X  e.  V  /\  u  C_  X )  /\  v  e.  u
)  ->  ( X  e.  V  /\  v  e.  X ) )
12 elsn 3973 . . . . . . . . . . . 12  |-  ( w  e.  { v }  <-> 
w  =  v )
13 eleq1a 2544 . . . . . . . . . . . . . . 15  |-  ( v  e.  X  ->  (
w  =  v  ->  w  e.  X )
)
14 simpl 464 . . . . . . . . . . . . . . . 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 2484 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( x  =  v  /\  y  =  w )  ->  ( x  =  y  <-> 
v  =  w ) )
1716ifbid 3894 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( x  =  v  /\  y  =  w )  ->  if ( x  =  y ,  0 ,  1 )  =  if ( v  =  w ,  0 ,  1 ) )
18 0re 9661 . . . . . . . . . . . . . . . . . . . . . 22  |-  0  e.  RR
19 1re 9660 . . . . . . . . . . . . . . . . . . . . . 22  |-  1  e.  RR
2018, 19keepel 3939 . . . . . . . . . . . . . . . . . . . . 21  |-  if ( v  =  w ,  0 ,  1 )  e.  RR
2120elexi 3041 . . . . . . . . . . . . . . . . . . . 20  |-  if ( v  =  w ,  0 ,  1 )  e.  _V
2217, 1, 21ovmpt2a 6446 . . . . . . . . . . . . . . . . . . 19  |-  ( ( v  e.  X  /\  w  e.  X )  ->  ( v D w )  =  if ( v  =  w ,  0 ,  1 ) )
2322breq1d 4405 . . . . . . . . . . . . . . . . . 18  |-  ( ( v  e.  X  /\  w  e.  X )  ->  ( ( v D w )  <  1  <->  if ( v  =  w ,  0 ,  1 )  <  1 ) )
2419ltnri 9761 . . . . . . . . . . . . . . . . . . . . . 22  |-  -.  1  <  1
25 iffalse 3881 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( -.  v  =  w  ->  if ( v  =  w ,  0 ,  1 )  =  1 )
2625breq1d 4405 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( -.  v  =  w  -> 
( if ( v  =  w ,  0 ,  1 )  <  1  <->  1  <  1
) )
2724, 26mtbiri 310 . . . . . . . . . . . . . . . . . . . . 21  |-  ( -.  v  =  w  ->  -.  if ( v  =  w ,  0 ,  1 )  <  1
)
2827con4i 135 . . . . . . . . . . . . . . . . . . . 20  |-  ( if ( v  =  w ,  0 ,  1 )  <  1  -> 
v  =  w )
29 iftrue 3878 . . . . . . . . . . . . . . . . . . . . 21  |-  ( v  =  w  ->  if ( v  =  w ,  0 ,  1 )  =  0 )
30 0lt1 10157 . . . . . . . . . . . . . . . . . . . . 21  |-  0  <  1
3129, 30syl6eqbr 4433 . . . . . . . . . . . . . . . . . . . 20  |-  ( v  =  w  ->  if ( v  =  w ,  0 ,  1 )  <  1 )
3228, 31impbii 192 . . . . . . . . . . . . . . . . . . 19  |-  ( if ( v  =  w ,  0 ,  1 )  <  1  <->  v  =  w )
33 equcom 1870 . . . . . . . . . . . . . . . . . . 19  |-  ( v  =  w  <->  w  =  v )
3432, 33bitri 257 . . . . . . . . . . . . . . . . . 18  |-  ( if ( v  =  w ,  0 ,  1 )  <  1  <->  w  =  v )
3523, 34syl6rbb 270 . . . . . . . . . . . . . . . . 17  |-  ( ( v  e.  X  /\  w  e.  X )  ->  ( w  =  v  <-> 
( v D w )  <  1 ) )
36 simpr 468 . . . . . . . . . . . . . . . . . 18  |-  ( ( v  e.  X  /\  w  e.  X )  ->  w  e.  X )
3736biantrurd 516 . . . . . . . . . . . . . . . . 17  |-  ( ( v  e.  X  /\  w  e.  X )  ->  ( ( v D w )  <  1  <->  ( w  e.  X  /\  ( v D w )  <  1 ) ) )
3835, 37bitrd 261 . . . . . . . . . . . . . . . 16  |-  ( ( v  e.  X  /\  w  e.  X )  ->  ( w  =  v  <-> 
( w  e.  X  /\  ( v D w )  <  1 ) ) )
3938ex 441 . . . . . . . . . . . . . . 15  |-  ( v  e.  X  ->  (
w  e.  X  -> 
( w  =  v  <-> 
( w  e.  X  /\  ( v D w )  <  1 ) ) ) )
4013, 15, 39pm5.21ndd 361 . . . . . . . . . . . . . 14  |-  ( v  e.  X  ->  (
w  =  v  <->  ( w  e.  X  /\  (
v D w )  <  1 ) ) )
4140adantl 473 . . . . . . . . . . . . 13  |-  ( ( X  e.  V  /\  v  e.  X )  ->  ( w  =  v  <-> 
( w  e.  X  /\  ( v D w )  <  1 ) ) )
42 1rp 11329 . . . . . . . . . . . . . . . 16  |-  1  e.  RR+
43 rpxr 11332 . . . . . . . . . . . . . . . 16  |-  ( 1  e.  RR+  ->  1  e. 
RR* )
4442, 43ax-mp 5 . . . . . . . . . . . . . . 15  |-  1  e.  RR*
45 elbl 21481 . . . . . . . . . . . . . . 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 1379 . . . . . . . . . . . . . 14  |-  ( ( D  e.  ( *Met `  X )  /\  v  e.  X
)  ->  ( w  e.  ( v ( ball `  D ) 1 )  <-> 
( w  e.  X  /\  ( v D w )  <  1 ) ) )
474, 46sylan 479 . . . . . . . . . . . . 13  |-  ( ( X  e.  V  /\  v  e.  X )  ->  ( w  e.  ( v ( ball `  D
) 1 )  <->  ( w  e.  X  /\  (
v D w )  <  1 ) ) )
4841, 47bitr4d 264 . . . . . . . . . . . 12  |-  ( ( X  e.  V  /\  v  e.  X )  ->  ( w  =  v  <-> 
w  e.  ( v ( ball `  D
) 1 ) ) )
4912, 48syl5bb 265 . . . . . . . . . . 11  |-  ( ( X  e.  V  /\  v  e.  X )  ->  ( w  e.  {
v }  <->  w  e.  ( v ( ball `  D ) 1 ) ) )
5049eqrdv 2469 . . . . . . . . . 10  |-  ( ( X  e.  V  /\  v  e.  X )  ->  { v }  =  ( v ( ball `  D ) 1 ) )
51 blelrn 21510 . . . . . . . . . . . 12  |-  ( ( D  e.  ( *Met `  X )  /\  v  e.  X  /\  1  e.  RR* )  ->  ( v ( ball `  D ) 1 )  e.  ran  ( ball `  D ) )
5244, 51mp3an3 1379 . . . . . . . . . . 11  |-  ( ( D  e.  ( *Met `  X )  /\  v  e.  X
)  ->  ( v
( ball `  D )
1 )  e.  ran  ( ball `  D )
)
534, 52sylan 479 . . . . . . . . . 10  |-  ( ( X  e.  V  /\  v  e.  X )  ->  ( v ( ball `  D ) 1 )  e.  ran  ( ball `  D ) )
5450, 53eqeltrd 2549 . . . . . . . . 9  |-  ( ( X  e.  V  /\  v  e.  X )  ->  { v }  e.  ran  ( ball `  D
) )
55 snssi 4107 . . . . . . . . . 10  |-  ( v  e.  u  ->  { v }  C_  u )
56 ssnid 3989 . . . . . . . . . 10  |-  v  e. 
{ v }
5755, 56jctil 546 . . . . . . . . 9  |-  ( v  e.  u  ->  (
v  e.  { v }  /\  { v }  C_  u )
)
58 eleq2 2538 . . . . . . . . . . 11  |-  ( w  =  { v }  ->  ( v  e.  w  <->  v  e.  {
v } ) )
59 sseq1 3439 . . . . . . . . . . 11  |-  ( w  =  { v }  ->  ( w  C_  u 
<->  { v }  C_  u ) )
6058, 59anbi12d 725 . . . . . . . . . 10  |-  ( w  =  { v }  ->  ( ( v  e.  w  /\  w  C_  u )  <->  ( v  e.  { v }  /\  { v }  C_  u
) ) )
6160rspcev 3136 . . . . . . . . 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 485 . . . . . . . 8  |-  ( ( ( X  e.  V  /\  v  e.  X
)  /\  v  e.  u )  ->  E. w  e.  ran  ( ball `  D
) ( v  e.  w  /\  w  C_  u ) )
6311, 62sylancom 680 . . . . . . 7  |-  ( ( ( X  e.  V  /\  u  C_  X )  /\  v  e.  u
)  ->  E. w  e.  ran  ( ball `  D
) ( v  e.  w  /\  w  C_  u ) )
6463ralrimiva 2809 . . . . . 6  |-  ( ( X  e.  V  /\  u  C_  X )  ->  A. v  e.  u  E. w  e.  ran  ( ball `  D )
( v  e.  w  /\  w  C_  u ) )
6564ex 441 . . . . 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 646 . . . 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 264 . . 3  |-  ( X  e.  V  ->  (
u  e.  ( MetOpen `  D )  <->  u  C_  X
) )
68 selpw 3949 . . 3  |-  ( u  e.  ~P X  <->  u  C_  X
)
6967, 68syl6bbr 271 . 2  |-  ( X  e.  V  ->  (
u  e.  ( MetOpen `  D )  <->  u  e.  ~P X ) )
7069eqrdv 2469 1  |-  ( X  e.  V  ->  ( MetOpen
`  D )  =  ~P X )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    /\ wa 376    = wceq 1452    e. wcel 1904   A.wral 2756   E.wrex 2757    C_ wss 3390   ifcif 3872   ~Pcpw 3942   {csn 3959   class class class wbr 4395   ran crn 4840   ` cfv 5589  (class class class)co 6308    |-> cmpt2 6310   RRcr 9556   0cc0 9557   1c1 9558   RR*cxr 9692    < clt 9693   RR+crp 11325   *Metcxmt 19032   Metcme 19033   ballcbl 19034   MetOpencmopn 19037
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-er 7381  df-map 7492  df-en 7588  df-dom 7589  df-sdom 7590  df-sup 7974  df-inf 7975  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-n0 10894  df-z 10962  df-uz 11183  df-q 11288  df-rp 11326  df-xneg 11432  df-xadd 11433  df-xmul 11434  df-topgen 15420  df-psmet 19039  df-xmet 19040  df-met 19041  df-bl 19042  df-mopn 19043  df-bases 19999
This theorem is referenced by: (None)
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