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Theorem dscopn 21220
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 21219 . . . . . 6  |-  ( X  e.  V  ->  D  e.  ( Met `  X
) )
3 metxmet 20963 . . . . . 6  |-  ( D  e.  ( Met `  X
)  ->  D  e.  ( *Met `  X
) )
42, 3syl 16 . . . . 5  |-  ( X  e.  V  ->  D  e.  ( *Met `  X ) )
5 eqid 2457 . . . . . 6  |-  ( MetOpen `  D )  =  (
MetOpen `  D )
65elmopn 21071 . . . . 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 3494 . . . . . . . . . 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 4046 . . . . . . . . . . . 12  |-  ( w  e.  { v }  <-> 
w  =  v )
13 eleq1a 2540 . . . . . . . . . . . . . . 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 2476 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( x  =  v  /\  y  =  w )  ->  ( x  =  y  <-> 
v  =  w ) )
1716ifbid 3966 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( x  =  v  /\  y  =  w )  ->  if ( x  =  y ,  0 ,  1 )  =  if ( v  =  w ,  0 ,  1 ) )
18 0re 9613 . . . . . . . . . . . . . . . . . . . . . 22  |-  0  e.  RR
19 1re 9612 . . . . . . . . . . . . . . . . . . . . . 22  |-  1  e.  RR
2018, 19keepel 4012 . . . . . . . . . . . . . . . . . . . . 21  |-  if ( v  =  w ,  0 ,  1 )  e.  RR
2120elexi 3119 . . . . . . . . . . . . . . . . . . . 20  |-  if ( v  =  w ,  0 ,  1 )  e.  _V
2217, 1, 21ovmpt2a 6432 . . . . . . . . . . . . . . . . . . 19  |-  ( ( v  e.  X  /\  w  e.  X )  ->  ( v D w )  =  if ( v  =  w ,  0 ,  1 ) )
2322breq1d 4466 . . . . . . . . . . . . . . . . . 18  |-  ( ( v  e.  X  /\  w  e.  X )  ->  ( ( v D w )  <  1  <->  if ( v  =  w ,  0 ,  1 )  <  1 ) )
2419ltnri 9710 . . . . . . . . . . . . . . . . . . . . . 22  |-  -.  1  <  1
25 iffalse 3953 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( -.  v  =  w  ->  if ( v  =  w ,  0 ,  1 )  =  1 )
2625breq1d 4466 . . . . . . . . . . . . . . . . . . . . . 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 3950 . . . . . . . . . . . . . . . . . . . . 21  |-  ( v  =  w  ->  if ( v  =  w ,  0 ,  1 )  =  0 )
30 0lt1 10096 . . . . . . . . . . . . . . . . . . . . 21  |-  0  <  1
3129, 30syl6eqbr 4493 . . . . . . . . . . . . . . . . . . . 20  |-  ( v  =  w  ->  if ( v  =  w ,  0 ,  1 )  <  1 )
3228, 31impbii 188 . . . . . . . . . . . . . . . . . . 19  |-  ( if ( v  =  w ,  0 ,  1 )  <  1  <->  v  =  w )
33 equcom 1795 . . . . . . . . . . . . . . . . . . 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 11249 . . . . . . . . . . . . . . . 16  |-  1  e.  RR+
43 rpxr 11252 . . . . . . . . . . . . . . . 16  |-  ( 1  e.  RR+  ->  1  e. 
RR* )
4442, 43ax-mp 5 . . . . . . . . . . . . . . 15  |-  1  e.  RR*
45 elbl 21017 . . . . . . . . . . . . . . 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 2454 . . . . . . . . . 10  |-  ( ( X  e.  V  /\  v  e.  X )  ->  { v }  =  ( v ( ball `  D ) 1 ) )
51 blelrn 21046 . . . . . . . . . . . 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 2545 . . . . . . . . 9  |-  ( ( X  e.  V  /\  v  e.  X )  ->  { v }  e.  ran  ( ball `  D
) )
55 snssi 4176 . . . . . . . . . 10  |-  ( v  e.  u  ->  { v }  C_  u )
56 ssnid 4061 . . . . . . . . . 10  |-  v  e. 
{ v }
5755, 56jctil 537 . . . . . . . . 9  |-  ( v  e.  u  ->  (
v  e.  { v }  /\  { v }  C_  u )
)
58 eleq2 2530 . . . . . . . . . . 11  |-  ( w  =  { v }  ->  ( v  e.  w  <->  v  e.  {
v } ) )
59 sseq1 3520 . . . . . . . . . . 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 3210 . . . . . . . . 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 2871 . . . . . 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 4022 . . 3  |-  ( u  e.  ~P X  <->  u  C_  X
)
6967, 68syl6bbr 263 . 2  |-  ( X  e.  V  ->  (
u  e.  ( MetOpen `  D )  <->  u  e.  ~P X ) )
7069eqrdv 2454 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 1395    e. wcel 1819   A.wral 2807   E.wrex 2808    C_ wss 3471   ifcif 3944   ~Pcpw 4015   {csn 4032   class class class wbr 4456   ran crn 5009   ` cfv 5594  (class class class)co 6296    |-> cmpt2 6298   RRcr 9508   0cc0 9509   1c1 9510   RR*cxr 9644    < clt 9645   RR+crp 11245   *Metcxmt 18530   Metcme 18531   ballcbl 18532   MetOpencmopn 18535
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  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 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-sup 7919  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-n0 10817  df-z 10886  df-uz 11107  df-q 11208  df-rp 11246  df-xneg 11343  df-xadd 11344  df-xmul 11345  df-topgen 14861  df-psmet 18538  df-xmet 18539  df-met 18540  df-bl 18541  df-mopn 18542  df-bases 19528
This theorem is referenced by: (None)
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