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Theorem isms 13586
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 5510 . . . . 5  |-  ( f  =  K  ->  ( dist `  f )  =  ( dist `  K
) )
2 fveq2 5510 . . . . . . 7  |-  ( f  =  K  ->  ( Base `  f )  =  ( Base `  K
) )
3 isms.x . . . . . . 7  |-  X  =  ( Base `  K
)
42, 3eqtr4di 2228 . . . . . 6  |-  ( f  =  K  ->  ( Base `  f )  =  X )
54sqxpeqd 4648 . . . . 5  |-  ( f  =  K  ->  (
( Base `  f )  X.  ( Base `  f
) )  =  ( X  X.  X ) )
61, 5reseq12d 4903 . . . 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, 7eqtr4di 2228 . . 3  |-  ( f  =  K  ->  (
( dist `  f )  |`  ( ( Base `  f
)  X.  ( Base `  f ) ) )  =  D )
94fveq2d 5514 . . 3  |-  ( f  =  K  ->  ( Met `  ( Base `  f
) )  =  ( Met `  X ) )
108, 9eleq12d 2248 . 2  |-  ( f  =  K  ->  (
( ( dist `  f
)  |`  ( ( Base `  f )  X.  ( Base `  f ) ) )  e.  ( Met `  ( Base `  f
) )  <->  D  e.  ( Met `  X ) ) )
11 df-ms 13473 . 2  |-  MetSp  =  {
f  e.  *MetSp  |  ( ( dist `  f
)  |`  ( ( Base `  f )  X.  ( Base `  f ) ) )  e.  ( Met `  ( Base `  f
) ) }
1210, 11elrab2 2896 1  |-  ( K  e.  MetSp 
<->  ( K  e.  *MetSp  /\  D  e.  ( Met `  X ) ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 104    <-> wb 105    = wceq 1353    e. wcel 2148    X. cxp 4620    |` cres 4624   ` cfv 5211   Basecbs 12432   distcds 12514   TopOpenctopn 12624   Metcmet 13114   *MetSpcxms 13469   MetSpcms 13470
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-ext 2159
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-rex 2461  df-rab 2464  df-v 2739  df-un 3133  df-in 3135  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-xp 4628  df-res 4634  df-iota 5173  df-fv 5219  df-ms 13473
This theorem is referenced by:  isms2  13587  msxms  13591  mspropd  13611
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