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Theorem isms 18436
Description: Express the predicate " <. X ,  D >. is a metric space" with underlying set  X and distance function  D. (Contributed by NM, 27-Aug-2006.) (Revised by Mario Carneiro, 24-Aug-2015.)
Hypotheses
Ref Expression
isms.j  |-  J  =  ( TopOpen `  K )
isms.x  |-  X  =  ( Base `  K
)
isms.d  |-  D  =  ( ( dist `  K
)  |`  ( X  X.  X ) )
Assertion
Ref Expression
isms  |-  ( K  e.  MetSp 
<->  ( K  e.  * MetSp  /\  D  e.  ( Met `  X ) ) )

Proof of Theorem isms
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 fveq2 5691 . . . . 5  |-  ( f  =  K  ->  ( dist `  f )  =  ( dist `  K
) )
2 fveq2 5691 . . . . . . 7  |-  ( f  =  K  ->  ( Base `  f )  =  ( Base `  K
) )
3 isms.x . . . . . . 7  |-  X  =  ( Base `  K
)
42, 3syl6eqr 2458 . . . . . 6  |-  ( f  =  K  ->  ( Base `  f )  =  X )
54, 4xpeq12d 4866 . . . . 5  |-  ( f  =  K  ->  (
( Base `  f )  X.  ( Base `  f
) )  =  ( X  X.  X ) )
61, 5reseq12d 5110 . . . 4  |-  ( f  =  K  ->  (
( dist `  f )  |`  ( ( Base `  f
)  X.  ( Base `  f ) ) )  =  ( ( dist `  K )  |`  ( X  X.  X ) ) )
7 isms.d . . . 4  |-  D  =  ( ( dist `  K
)  |`  ( X  X.  X ) )
86, 7syl6eqr 2458 . . 3  |-  ( f  =  K  ->  (
( dist `  f )  |`  ( ( Base `  f
)  X.  ( Base `  f ) ) )  =  D )
94fveq2d 5695 . . 3  |-  ( f  =  K  ->  ( Met `  ( Base `  f
) )  =  ( Met `  X ) )
108, 9eleq12d 2476 . 2  |-  ( f  =  K  ->  (
( ( dist `  f
)  |`  ( ( Base `  f )  X.  ( Base `  f ) ) )  e.  ( Met `  ( Base `  f
) )  <->  D  e.  ( Met `  X ) ) )
11 df-ms 18308 . 2  |-  MetSp  =  {
f  e.  * MetSp  |  ( ( dist `  f
)  |`  ( ( Base `  f )  X.  ( Base `  f ) ) )  e.  ( Met `  ( Base `  f
) ) }
1210, 11elrab2 3058 1  |-  ( K  e.  MetSp 
<->  ( K  e.  * MetSp  /\  D  e.  ( Met `  X ) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721    X. cxp 4839    |` cres 4843   ` cfv 5417   Basecbs 13428   distcds 13497   TopOpenctopn 13608   Metcme 16646   *
MetSpcxme 18304   MetSpcmt 18305
This theorem is referenced by:  isms2  18437  msxms  18441  mspropd  18461  setsms  18467  tmsms  18474  imasf1oms  18477  ressms  18513  prdsms  18518
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-rex 2676  df-rab 2679  df-v 2922  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-br 4177  df-opab 4231  df-xp 4847  df-res 4853  df-iota 5381  df-fv 5425  df-ms 18308
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