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Theorem isms 20680
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 5857 . . . . 5  |-  ( f  =  K  ->  ( dist `  f )  =  ( dist `  K
) )
2 fveq2 5857 . . . . . . 7  |-  ( f  =  K  ->  ( Base `  f )  =  ( Base `  K
) )
3 isms.x . . . . . . 7  |-  X  =  ( Base `  K
)
42, 3syl6eqr 2519 . . . . . 6  |-  ( f  =  K  ->  ( Base `  f )  =  X )
54, 4xpeq12d 5017 . . . . 5  |-  ( f  =  K  ->  (
( Base `  f )  X.  ( Base `  f
) )  =  ( X  X.  X ) )
61, 5reseq12d 5265 . . . 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 2519 . . 3  |-  ( f  =  K  ->  (
( dist `  f )  |`  ( ( Base `  f
)  X.  ( Base `  f ) ) )  =  D )
94fveq2d 5861 . . 3  |-  ( f  =  K  ->  ( Met `  ( Base `  f
) )  =  ( Met `  X ) )
108, 9eleq12d 2542 . 2  |-  ( f  =  K  ->  (
( ( dist `  f
)  |`  ( ( Base `  f )  X.  ( Base `  f ) ) )  e.  ( Met `  ( Base `  f
) )  <->  D  e.  ( Met `  X ) ) )
11 df-ms 20552 . 2  |-  MetSp  =  {
f  e.  *MetSp  |  ( ( dist `  f
)  |`  ( ( Base `  f )  X.  ( Base `  f ) ) )  e.  ( Met `  ( Base `  f
) ) }
1210, 11elrab2 3256 1  |-  ( K  e.  MetSp 
<->  ( K  e.  *MetSp  /\  D  e.  ( Met `  X ) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1374    e. wcel 1762    X. cxp 4990    |` cres 4994   ` cfv 5579   Basecbs 14479   distcds 14553   TopOpenctopn 14666   Metcme 18168   *MetSpcxme 20548   MetSpcmt 20549
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-rex 2813  df-rab 2816  df-v 3108  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-xp 4998  df-res 5004  df-iota 5542  df-fv 5587  df-ms 20552
This theorem is referenced by:  isms2  20681  msxms  20685  mspropd  20705  setsms  20711  tmsms  20718  imasf1oms  20721  ressms  20757  prdsms  20762
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