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Theorem metidval 26269
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 26267 . . 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 5037 . . . . . . . 8  |-  ( ( D  e.  (PsMet `  X )  /\  d  =  D )  ->  dom  d  =  dom  D )
54dmeqd 5037 . . . . . . 7  |-  ( ( D  e.  (PsMet `  X )  /\  d  =  D )  ->  dom  dom  d  =  dom  dom  D )
6 psmetdmdm 19856 . . . . . . . 8  |-  ( D  e.  (PsMet `  X
)  ->  X  =  dom  dom  D )
76adantr 465 . . . . . . 7  |-  ( ( D  e.  (PsMet `  X )  /\  d  =  D )  ->  X  =  dom  dom  D )
85, 7eqtr4d 2473 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  d  =  D )  ->  dom  dom  d  =  X )
98eleq2d 2505 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  d  =  D )  ->  (
x  e.  dom  dom  d 
<->  x  e.  X ) )
108eleq2d 2505 . . . . 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 6103 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  d  =  D )  ->  (
x d y )  =  ( x D y ) )
1312eqeq1d 2446 . . . 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 4350 . 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 5711 . . . 4  |-  ( D  e.  (PsMet `  X
)  ->  X  e.  dom PsMet )
17 fveq2 5686 . . . . . 6  |-  ( x  =  X  ->  (PsMet `  x )  =  (PsMet `  X ) )
1817eleq2d 2505 . . . . 5  |-  ( x  =  X  ->  ( D  e.  (PsMet `  x
)  <->  D  e.  (PsMet `  X ) ) )
1918rspcev 3068 . . . 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 17784 . . . . 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 5450 . . . 4  |-  Fun PsMet
23 elunirn 5963 . . . 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 4906 . . 3  |-  { <. x ,  y >.  |  ( ( x  e.  X  /\  y  e.  X
)  /\  ( x D y )  =  0 ) }  C_  ( X  X.  X
)
27 elfvex 5712 . . . 4  |-  ( D  e.  (PsMet `  X
)  ->  X  e.  _V )
28 xpexg 6502 . . . 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 4433 . . 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 5774 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 1369    e. wcel 1756   A.wral 2710   E.wrex 2711   {crab 2714   _Vcvv 2967    C_ wss 3323   U.cuni 4086   class class class wbr 4287   {copab 4344    e. cmpt 4345    X. cxp 4833   dom cdm 4835   ran crn 4836   Fun wfun 5407   ` cfv 5413  (class class class)co 6086    ^m cmap 7206   0cc0 9274   RR*cxr 9409    <_ cle 9411   +ecxad 11079  PsMetcpsmet 17775  ~Metcmetid 26265
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-fv 5421  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-map 7208  df-xr 9414  df-psmet 17784  df-metid 26267
This theorem is referenced by:  metidss  26270  metidv  26271
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