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Theorem metrel 13845
Description: The class of metrics is a relation. (Contributed by Jim Kingdon, 20-Apr-2023.)
Assertion
Ref Expression
metrel  |-  Rel  Met

Proof of Theorem metrel
Dummy variables  e  d  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mptrel 4756 . 2  |-  Rel  (
e  e.  _V  |->  { d  e.  ( RR 
^m  ( e  X.  e ) )  | 
A. x  e.  e 
A. y  e.  e  ( ( ( x d y )  =  0  <->  x  =  y
)  /\  A. z  e.  e  ( x
d y )  <_ 
( ( z d x )  +  ( z d y ) ) ) } )
2 df-met 13452 . . 3  |-  Met  =  ( e  e.  _V  |->  { d  e.  ( RR  ^m  ( e  X.  e ) )  |  A. x  e.  e  A. y  e.  e  ( ( ( x d y )  =  0  <->  x  =  y )  /\  A. z  e.  e  (
x d y )  <_  ( ( z d x )  +  ( z d y ) ) ) } )
32releqi 4710 . 2  |-  ( Rel 
Met 
<->  Rel  ( e  e. 
_V  |->  { d  e.  ( RR  ^m  (
e  X.  e ) )  |  A. x  e.  e  A. y  e.  e  ( (
( x d y )  =  0  <->  x  =  y )  /\  A. z  e.  e  ( x d y )  <_  ( ( z d x )  +  ( z d y ) ) ) } ) )
41, 3mpbir 146 1  |-  Rel  Met
Colors of variables: wff set class
Syntax hints:    /\ wa 104    <-> wb 105    = wceq 1353   A.wral 2455   {crab 2459   _Vcvv 2738   class class class wbr 4004    |-> cmpt 4065    X. cxp 4625   Rel wrel 4632  (class class class)co 5875    ^m cmap 6648   RRcr 7810   0cc0 7811    + caddc 7814    <_ cle 7993   Metcmet 13444
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2740  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-opab 4066  df-mpt 4067  df-xp 4633  df-rel 4634  df-met 13452
This theorem is referenced by:  metflem  13852  ismet2  13857
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