Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  metidval Structured version   Unicode version

Theorem metidval 28335
Description: Value of the metric identification relation. (Contributed by Thierry Arnoux, 7-Feb-2018.)
Assertion
Ref Expression
metidval  |-  ( D  e.  (PsMet `  X
)  ->  (~Met `  D
)  =  { <. x ,  y >.  |  ( ( x  e.  X  /\  y  e.  X
)  /\  ( x D y )  =  0 ) } )
Distinct variable groups:    x, y, D    x, X, y

Proof of Theorem metidval
Dummy variables  w  d  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-metid 28333 . . 3  |- ~Met  =  ( d  e.  U. ran PsMet  |->  { <. x ,  y
>.  |  ( (
x  e.  dom  dom  d  /\  y  e.  dom  dom  d )  /\  (
x d y )  =  0 ) } )
21a1i 11 . 2  |-  ( D  e.  (PsMet `  X
)  -> ~Met  =  (
d  e.  U. ran PsMet  |->  { <. x ,  y
>.  |  ( (
x  e.  dom  dom  d  /\  y  e.  dom  dom  d )  /\  (
x d y )  =  0 ) } ) )
3 simpr 461 . . . . . . . . 9  |-  ( ( D  e.  (PsMet `  X )  /\  d  =  D )  ->  d  =  D )
43dmeqd 5028 . . . . . . . 8  |-  ( ( D  e.  (PsMet `  X )  /\  d  =  D )  ->  dom  d  =  dom  D )
54dmeqd 5028 . . . . . . 7  |-  ( ( D  e.  (PsMet `  X )  /\  d  =  D )  ->  dom  dom  d  =  dom  dom  D )
6 psmetdmdm 21103 . . . . . . . 8  |-  ( D  e.  (PsMet `  X
)  ->  X  =  dom  dom  D )
76adantr 465 . . . . . . 7  |-  ( ( D  e.  (PsMet `  X )  /\  d  =  D )  ->  X  =  dom  dom  D )
85, 7eqtr4d 2448 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  d  =  D )  ->  dom  dom  d  =  X )
98eleq2d 2474 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  d  =  D )  ->  (
x  e.  dom  dom  d 
<->  x  e.  X ) )
108eleq2d 2474 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  d  =  D )  ->  (
y  e.  dom  dom  d 
<->  y  e.  X ) )
119, 10anbi12d 711 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  d  =  D )  ->  (
( x  e.  dom  dom  d  /\  y  e. 
dom  dom  d )  <->  ( x  e.  X  /\  y  e.  X ) ) )
123oveqd 6297 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  d  =  D )  ->  (
x d y )  =  ( x D y ) )
1312eqeq1d 2406 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  d  =  D )  ->  (
( x d y )  =  0  <->  (
x D y )  =  0 ) )
1411, 13anbi12d 711 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  d  =  D )  ->  (
( ( x  e. 
dom  dom  d  /\  y  e.  dom  dom  d )  /\  ( x d y )  =  0 )  <-> 
( ( x  e.  X  /\  y  e.  X )  /\  (
x D y )  =  0 ) ) )
1514opabbidv 4460 . 2  |-  ( ( D  e.  (PsMet `  X )  /\  d  =  D )  ->  { <. x ,  y >.  |  ( ( x  e.  dom  dom  d  /\  y  e. 
dom  dom  d )  /\  ( x d y )  =  0 ) }  =  { <. x ,  y >.  |  ( ( x  e.  X  /\  y  e.  X
)  /\  ( x D y )  =  0 ) } )
16 elfvdm 5877 . . . 4  |-  ( D  e.  (PsMet `  X
)  ->  X  e.  dom PsMet )
17 fveq2 5851 . . . . . 6  |-  ( x  =  X  ->  (PsMet `  x )  =  (PsMet `  X ) )
1817eleq2d 2474 . . . . 5  |-  ( x  =  X  ->  ( D  e.  (PsMet `  x
)  <->  D  e.  (PsMet `  X ) ) )
1918rspcev 3162 . . . 4  |-  ( ( X  e.  dom PsMet  /\  D  e.  (PsMet `  X )
)  ->  E. x  e.  dom PsMet D  e.  (PsMet `  x ) )
2016, 19mpancom 669 . . 3  |-  ( D  e.  (PsMet `  X
)  ->  E. x  e.  dom PsMet D  e.  (PsMet `  x ) )
21 df-psmet 18733 . . . . 5  |- PsMet  =  ( x  e.  _V  |->  { d  e.  ( RR*  ^m  ( x  X.  x
) )  |  A. y  e.  x  (
( y d y )  =  0  /\ 
A. z  e.  x  A. w  e.  x  ( y d z )  <_  ( (
w d y ) +e ( w d z ) ) ) } )
2221funmpt2 5608 . . . 4  |-  Fun PsMet
23 elunirn 6146 . . . 4  |-  ( Fun PsMet  ->  ( D  e.  U. ran PsMet  <->  E. x  e.  dom PsMet D  e.  (PsMet `  x
) ) )
2422, 23ax-mp 5 . . 3  |-  ( D  e.  U. ran PsMet  <->  E. x  e.  dom PsMet D  e.  (PsMet `  x ) )
2520, 24sylibr 214 . 2  |-  ( D  e.  (PsMet `  X
)  ->  D  e.  U.
ran PsMet )
26 opabssxp 4900 . . 3  |-  { <. x ,  y >.  |  ( ( x  e.  X  /\  y  e.  X
)  /\  ( x D y )  =  0 ) }  C_  ( X  X.  X
)
27 elfvex 5878 . . . 4  |-  ( D  e.  (PsMet `  X
)  ->  X  e.  _V )
28 xpexg 6586 . . . 4  |-  ( ( X  e.  _V  /\  X  e.  _V )  ->  ( X  X.  X
)  e.  _V )
2927, 27, 28syl2anc 661 . . 3  |-  ( D  e.  (PsMet `  X
)  ->  ( X  X.  X )  e.  _V )
30 ssexg 4542 . . 3  |-  ( ( { <. x ,  y
>.  |  ( (
x  e.  X  /\  y  e.  X )  /\  ( x D y )  =  0 ) }  C_  ( X  X.  X )  /\  ( X  X.  X )  e. 
_V )  ->  { <. x ,  y >.  |  ( ( x  e.  X  /\  y  e.  X
)  /\  ( x D y )  =  0 ) }  e.  _V )
3126, 29, 30sylancr 663 . 2  |-  ( D  e.  (PsMet `  X
)  ->  { <. x ,  y >.  |  ( ( x  e.  X  /\  y  e.  X
)  /\  ( x D y )  =  0 ) }  e.  _V )
322, 15, 25, 31fvmptd 5940 1  |-  ( D  e.  (PsMet `  X
)  ->  (~Met `  D
)  =  { <. x ,  y >.  |  ( ( x  e.  X  /\  y  e.  X
)  /\  ( x D y )  =  0 ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 186    /\ wa 369    = wceq 1407    e. wcel 1844   A.wral 2756   E.wrex 2757   {crab 2760   _Vcvv 3061    C_ wss 3416   U.cuni 4193   class class class wbr 4397   {copab 4454    |-> cmpt 4455    X. cxp 4823   dom cdm 4825   ran crn 4826   Fun wfun 5565   ` cfv 5571  (class class class)co 6280    ^m cmap 7459   0cc0 9524   RR*cxr 9659    <_ cle 9661   +ecxad 11371  PsMetcpsmet 18724  ~Metcmetid 28331
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576  ax-cnex 9580  ax-resscn 9581
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-br 4398  df-opab 4456  df-mpt 4457  df-id 4740  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-fv 5579  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-map 7461  df-xr 9664  df-psmet 18733  df-metid 28333
This theorem is referenced by:  metidss  28336  metidv  28337
  Copyright terms: Public domain W3C validator