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Theorem metidval 26455
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 26453 . . 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 5143 . . . . . . . 8  |-  ( ( D  e.  (PsMet `  X )  /\  d  =  D )  ->  dom  d  =  dom  D )
54dmeqd 5143 . . . . . . 7  |-  ( ( D  e.  (PsMet `  X )  /\  d  =  D )  ->  dom  dom  d  =  dom  dom  D )
6 psmetdmdm 20006 . . . . . . . 8  |-  ( D  e.  (PsMet `  X
)  ->  X  =  dom  dom  D )
76adantr 465 . . . . . . 7  |-  ( ( D  e.  (PsMet `  X )  /\  d  =  D )  ->  X  =  dom  dom  D )
85, 7eqtr4d 2495 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  d  =  D )  ->  dom  dom  d  =  X )
98eleq2d 2521 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  d  =  D )  ->  (
x  e.  dom  dom  d 
<->  x  e.  X ) )
108eleq2d 2521 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  d  =  D )  ->  (
y  e.  dom  dom  d 
<->  y  e.  X ) )
119, 10anbi12d 710 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  d  =  D )  ->  (
( x  e.  dom  dom  d  /\  y  e. 
dom  dom  d )  <->  ( x  e.  X  /\  y  e.  X ) ) )
123oveqd 6210 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  d  =  D )  ->  (
x d y )  =  ( x D y ) )
1312eqeq1d 2453 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  d  =  D )  ->  (
( x d y )  =  0  <->  (
x D y )  =  0 ) )
1411, 13anbi12d 710 . . 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 4456 . 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 5818 . . . 4  |-  ( D  e.  (PsMet `  X
)  ->  X  e.  dom PsMet )
17 fveq2 5792 . . . . . 6  |-  ( x  =  X  ->  (PsMet `  x )  =  (PsMet `  X ) )
1817eleq2d 2521 . . . . 5  |-  ( x  =  X  ->  ( D  e.  (PsMet `  x
)  <->  D  e.  (PsMet `  X ) ) )
1918rspcev 3172 . . . 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 17927 . . . . 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 5556 . . . 4  |-  Fun PsMet
23 elunirn 6070 . . . 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 212 . 2  |-  ( D  e.  (PsMet `  X
)  ->  D  e.  U.
ran PsMet )
26 opabssxp 5012 . . 3  |-  { <. x ,  y >.  |  ( ( x  e.  X  /\  y  e.  X
)  /\  ( x D y )  =  0 ) }  C_  ( X  X.  X
)
27 elfvex 5819 . . . 4  |-  ( D  e.  (PsMet `  X
)  ->  X  e.  _V )
28 xpexg 6610 . . . 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 4539 . . 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 5881 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 184    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2795   E.wrex 2796   {crab 2799   _Vcvv 3071    C_ wss 3429   U.cuni 4192   class class class wbr 4393   {copab 4450    |-> cmpt 4451    X. cxp 4939   dom cdm 4941   ran crn 4942   Fun wfun 5513   ` cfv 5519  (class class class)co 6193    ^m cmap 7317   0cc0 9386   RR*cxr 9521    <_ cle 9523   +ecxad 11191  PsMetcpsmet 17918  ~Metcmetid 26451
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-cnex 9442  ax-resscn 9443
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-fv 5527  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-map 7319  df-xr 9526  df-psmet 17927  df-metid 26453
This theorem is referenced by:  metidss  26456  metidv  26457
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