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Theorem dscopn 21581
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 21580 . . . . . 6  |-  ( X  e.  V  ->  D  e.  ( Met `  X
) )
3 metxmet 21342 . . . . . 6  |-  ( D  e.  ( Met `  X
)  ->  D  e.  ( *Met `  X
) )
42, 3syl 17 . . . . 5  |-  ( X  e.  V  ->  D  e.  ( *Met `  X ) )
5 eqid 2450 . . . . . 6  |-  ( MetOpen `  D )  =  (
MetOpen `  D )
65elmopn 21450 . . . . 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 759 . . . . . . . . 9  |-  ( ( ( X  e.  V  /\  u  C_  X )  /\  v  e.  u
)  ->  X  e.  V )
9 ssel2 3426 . . . . . . . . . 10  |-  ( ( u  C_  X  /\  v  e.  u )  ->  v  e.  X )
109adantll 719 . . . . . . . . 9  |-  ( ( ( X  e.  V  /\  u  C_  X )  /\  v  e.  u
)  ->  v  e.  X )
118, 10jca 535 . . . . . . . 8  |-  ( ( ( X  e.  V  /\  u  C_  X )  /\  v  e.  u
)  ->  ( X  e.  V  /\  v  e.  X ) )
12 elsn 3981 . . . . . . . . . . . 12  |-  ( w  e.  { v }  <-> 
w  =  v )
13 eleq1a 2523 . . . . . . . . . . . . . . 15  |-  ( v  e.  X  ->  (
w  =  v  ->  w  e.  X )
)
14 simpl 459 . . . . . . . . . . . . . . . 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 2463 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( x  =  v  /\  y  =  w )  ->  ( x  =  y  <-> 
v  =  w ) )
1716ifbid 3902 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( x  =  v  /\  y  =  w )  ->  if ( x  =  y ,  0 ,  1 )  =  if ( v  =  w ,  0 ,  1 ) )
18 0re 9640 . . . . . . . . . . . . . . . . . . . . . 22  |-  0  e.  RR
19 1re 9639 . . . . . . . . . . . . . . . . . . . . . 22  |-  1  e.  RR
2018, 19keepel 3947 . . . . . . . . . . . . . . . . . . . . 21  |-  if ( v  =  w ,  0 ,  1 )  e.  RR
2120elexi 3054 . . . . . . . . . . . . . . . . . . . 20  |-  if ( v  =  w ,  0 ,  1 )  e.  _V
2217, 1, 21ovmpt2a 6424 . . . . . . . . . . . . . . . . . . 19  |-  ( ( v  e.  X  /\  w  e.  X )  ->  ( v D w )  =  if ( v  =  w ,  0 ,  1 ) )
2322breq1d 4411 . . . . . . . . . . . . . . . . . 18  |-  ( ( v  e.  X  /\  w  e.  X )  ->  ( ( v D w )  <  1  <->  if ( v  =  w ,  0 ,  1 )  <  1 ) )
2419ltnri 9740 . . . . . . . . . . . . . . . . . . . . . 22  |-  -.  1  <  1
25 iffalse 3889 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( -.  v  =  w  ->  if ( v  =  w ,  0 ,  1 )  =  1 )
2625breq1d 4411 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( -.  v  =  w  -> 
( if ( v  =  w ,  0 ,  1 )  <  1  <->  1  <  1
) )
2724, 26mtbiri 305 . . . . . . . . . . . . . . . . . . . . 21  |-  ( -.  v  =  w  ->  -.  if ( v  =  w ,  0 ,  1 )  <  1
)
2827con4i 134 . . . . . . . . . . . . . . . . . . . 20  |-  ( if ( v  =  w ,  0 ,  1 )  <  1  -> 
v  =  w )
29 iftrue 3886 . . . . . . . . . . . . . . . . . . . . 21  |-  ( v  =  w  ->  if ( v  =  w ,  0 ,  1 )  =  0 )
30 0lt1 10133 . . . . . . . . . . . . . . . . . . . . 21  |-  0  <  1
3129, 30syl6eqbr 4439 . . . . . . . . . . . . . . . . . . . 20  |-  ( v  =  w  ->  if ( v  =  w ,  0 ,  1 )  <  1 )
3228, 31impbii 191 . . . . . . . . . . . . . . . . . . 19  |-  ( if ( v  =  w ,  0 ,  1 )  <  1  <->  v  =  w )
33 equcom 1861 . . . . . . . . . . . . . . . . . . 19  |-  ( v  =  w  <->  w  =  v )
3432, 33bitri 253 . . . . . . . . . . . . . . . . . 18  |-  ( if ( v  =  w ,  0 ,  1 )  <  1  <->  w  =  v )
3523, 34syl6rbb 266 . . . . . . . . . . . . . . . . 17  |-  ( ( v  e.  X  /\  w  e.  X )  ->  ( w  =  v  <-> 
( v D w )  <  1 ) )
36 simpr 463 . . . . . . . . . . . . . . . . . 18  |-  ( ( v  e.  X  /\  w  e.  X )  ->  w  e.  X )
3736biantrurd 511 . . . . . . . . . . . . . . . . 17  |-  ( ( v  e.  X  /\  w  e.  X )  ->  ( ( v D w )  <  1  <->  ( w  e.  X  /\  ( v D w )  <  1 ) ) )
3835, 37bitrd 257 . . . . . . . . . . . . . . . 16  |-  ( ( v  e.  X  /\  w  e.  X )  ->  ( w  =  v  <-> 
( w  e.  X  /\  ( v D w )  <  1 ) ) )
3938ex 436 . . . . . . . . . . . . . . 15  |-  ( v  e.  X  ->  (
w  e.  X  -> 
( w  =  v  <-> 
( w  e.  X  /\  ( v D w )  <  1 ) ) ) )
4013, 15, 39pm5.21ndd 356 . . . . . . . . . . . . . 14  |-  ( v  e.  X  ->  (
w  =  v  <->  ( w  e.  X  /\  (
v D w )  <  1 ) ) )
4140adantl 468 . . . . . . . . . . . . 13  |-  ( ( X  e.  V  /\  v  e.  X )  ->  ( w  =  v  <-> 
( w  e.  X  /\  ( v D w )  <  1 ) ) )
42 1rp 11303 . . . . . . . . . . . . . . . 16  |-  1  e.  RR+
43 rpxr 11306 . . . . . . . . . . . . . . . 16  |-  ( 1  e.  RR+  ->  1  e. 
RR* )
4442, 43ax-mp 5 . . . . . . . . . . . . . . 15  |-  1  e.  RR*
45 elbl 21396 . . . . . . . . . . . . . . 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 1352 . . . . . . . . . . . . . 14  |-  ( ( D  e.  ( *Met `  X )  /\  v  e.  X
)  ->  ( w  e.  ( v ( ball `  D ) 1 )  <-> 
( w  e.  X  /\  ( v D w )  <  1 ) ) )
474, 46sylan 474 . . . . . . . . . . . . 13  |-  ( ( X  e.  V  /\  v  e.  X )  ->  ( w  e.  ( v ( ball `  D
) 1 )  <->  ( w  e.  X  /\  (
v D w )  <  1 ) ) )
4841, 47bitr4d 260 . . . . . . . . . . . 12  |-  ( ( X  e.  V  /\  v  e.  X )  ->  ( w  =  v  <-> 
w  e.  ( v ( ball `  D
) 1 ) ) )
4912, 48syl5bb 261 . . . . . . . . . . 11  |-  ( ( X  e.  V  /\  v  e.  X )  ->  ( w  e.  {
v }  <->  w  e.  ( v ( ball `  D ) 1 ) ) )
5049eqrdv 2448 . . . . . . . . . 10  |-  ( ( X  e.  V  /\  v  e.  X )  ->  { v }  =  ( v ( ball `  D ) 1 ) )
51 blelrn 21425 . . . . . . . . . . . 12  |-  ( ( D  e.  ( *Met `  X )  /\  v  e.  X  /\  1  e.  RR* )  ->  ( v ( ball `  D ) 1 )  e.  ran  ( ball `  D ) )
5244, 51mp3an3 1352 . . . . . . . . . . 11  |-  ( ( D  e.  ( *Met `  X )  /\  v  e.  X
)  ->  ( v
( ball `  D )
1 )  e.  ran  ( ball `  D )
)
534, 52sylan 474 . . . . . . . . . 10  |-  ( ( X  e.  V  /\  v  e.  X )  ->  ( v ( ball `  D ) 1 )  e.  ran  ( ball `  D ) )
5450, 53eqeltrd 2528 . . . . . . . . 9  |-  ( ( X  e.  V  /\  v  e.  X )  ->  { v }  e.  ran  ( ball `  D
) )
55 snssi 4115 . . . . . . . . . 10  |-  ( v  e.  u  ->  { v }  C_  u )
56 ssnid 3996 . . . . . . . . . 10  |-  v  e. 
{ v }
5755, 56jctil 540 . . . . . . . . 9  |-  ( v  e.  u  ->  (
v  e.  { v }  /\  { v }  C_  u )
)
58 eleq2 2517 . . . . . . . . . . 11  |-  ( w  =  { v }  ->  ( v  e.  w  <->  v  e.  {
v } ) )
59 sseq1 3452 . . . . . . . . . . 11  |-  ( w  =  { v }  ->  ( w  C_  u 
<->  { v }  C_  u ) )
6058, 59anbi12d 716 . . . . . . . . . 10  |-  ( w  =  { v }  ->  ( ( v  e.  w  /\  w  C_  u )  <->  ( v  e.  { v }  /\  { v }  C_  u
) ) )
6160rspcev 3149 . . . . . . . . 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 480 . . . . . . . 8  |-  ( ( ( X  e.  V  /\  v  e.  X
)  /\  v  e.  u )  ->  E. w  e.  ran  ( ball `  D
) ( v  e.  w  /\  w  C_  u ) )
6311, 62sylancom 672 . . . . . . 7  |-  ( ( ( X  e.  V  /\  u  C_  X )  /\  v  e.  u
)  ->  E. w  e.  ran  ( ball `  D
) ( v  e.  w  /\  w  C_  u ) )
6463ralrimiva 2801 . . . . . 6  |-  ( ( X  e.  V  /\  u  C_  X )  ->  A. v  e.  u  E. w  e.  ran  ( ball `  D )
( v  e.  w  /\  w  C_  u ) )
6564ex 436 . . . . 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 639 . . . 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 260 . . 3  |-  ( X  e.  V  ->  (
u  e.  ( MetOpen `  D )  <->  u  C_  X
) )
68 selpw 3957 . . 3  |-  ( u  e.  ~P X  <->  u  C_  X
)
6967, 68syl6bbr 267 . 2  |-  ( X  e.  V  ->  (
u  e.  ( MetOpen `  D )  <->  u  e.  ~P X ) )
7069eqrdv 2448 1  |-  ( X  e.  V  ->  ( MetOpen
`  D )  =  ~P X )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1443    e. wcel 1886   A.wral 2736   E.wrex 2737    C_ wss 3403   ifcif 3880   ~Pcpw 3950   {csn 3967   class class class wbr 4401   ran crn 4834   ` cfv 5581  (class class class)co 6288    |-> cmpt2 6290   RRcr 9535   0cc0 9536   1c1 9537   RR*cxr 9671    < clt 9672   RR+crp 11299   *Metcxmt 18948   Metcme 18949   ballcbl 18950   MetOpencmopn 18953
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613  ax-pre-sup 9614
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-om 6690  df-1st 6790  df-2nd 6791  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-er 7360  df-map 7471  df-en 7567  df-dom 7568  df-sdom 7569  df-sup 7953  df-inf 7954  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-div 10267  df-nn 10607  df-2 10665  df-n0 10867  df-z 10935  df-uz 11157  df-q 11262  df-rp 11300  df-xneg 11406  df-xadd 11407  df-xmul 11408  df-topgen 15335  df-psmet 18955  df-xmet 18956  df-met 18957  df-bl 18958  df-mopn 18959  df-bases 19915
This theorem is referenced by: (None)
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