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Theorem isms 20024
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 2493 . . . . . 6  |-  ( f  =  K  ->  ( Base `  f )  =  X )
54, 4xpeq12d 4865 . . . . 5  |-  ( f  =  K  ->  (
( Base `  f )  X.  ( Base `  f
) )  =  ( X  X.  X ) )
61, 5reseq12d 5111 . . . 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 2493 . . 3  |-  ( f  =  K  ->  (
( dist `  f )  |`  ( ( Base `  f
)  X.  ( Base `  f ) ) )  =  D )
94fveq2d 5695 . . 3  |-  ( f  =  K  ->  ( Met `  ( Base `  f
) )  =  ( Met `  X ) )
108, 9eleq12d 2511 . 2  |-  ( f  =  K  ->  (
( ( dist `  f
)  |`  ( ( Base `  f )  X.  ( Base `  f ) ) )  e.  ( Met `  ( Base `  f
) )  <->  D  e.  ( Met `  X ) ) )
11 df-ms 19896 . 2  |-  MetSp  =  {
f  e.  *MetSp  |  ( ( dist `  f
)  |`  ( ( Base `  f )  X.  ( Base `  f ) ) )  e.  ( Met `  ( Base `  f
) ) }
1210, 11elrab2 3119 1  |-  ( K  e.  MetSp 
<->  ( K  e.  *MetSp  /\  D  e.  ( Met `  X ) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    X. cxp 4838    |` cres 4842   ` cfv 5418   Basecbs 14174   distcds 14247   TopOpenctopn 14360   Metcme 17802   *MetSpcxme 19892   MetSpcmt 19893
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-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
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-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-rex 2721  df-rab 2724  df-v 2974  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-xp 4846  df-res 4852  df-iota 5381  df-fv 5426  df-ms 19896
This theorem is referenced by:  isms2  20025  msxms  20029  mspropd  20049  setsms  20055  tmsms  20062  imasf1oms  20065  ressms  20101  prdsms  20106
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