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Theorem isxms 21242
Description: Express the predicate " <. X ,  D >. is an extended metric space" with underlying set  X and distance function  D. (Contributed by Mario Carneiro, 2-Sep-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
isxms  |-  ( K  e.  *MetSp  <->  ( K  e.  TopSp  /\  J  =  ( MetOpen `  D )
) )

Proof of Theorem isxms
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 fveq2 5849 . . . 4  |-  ( f  =  K  ->  ( TopOpen
`  f )  =  ( TopOpen `  K )
)
2 isms.j . . . 4  |-  J  =  ( TopOpen `  K )
31, 2syl6eqr 2461 . . 3  |-  ( f  =  K  ->  ( TopOpen
`  f )  =  J )
4 fveq2 5849 . . . . . 6  |-  ( f  =  K  ->  ( dist `  f )  =  ( dist `  K
) )
5 fveq2 5849 . . . . . . . 8  |-  ( f  =  K  ->  ( Base `  f )  =  ( Base `  K
) )
6 isms.x . . . . . . . 8  |-  X  =  ( Base `  K
)
75, 6syl6eqr 2461 . . . . . . 7  |-  ( f  =  K  ->  ( Base `  f )  =  X )
87sqxpeqd 4849 . . . . . 6  |-  ( f  =  K  ->  (
( Base `  f )  X.  ( Base `  f
) )  =  ( X  X.  X ) )
94, 8reseq12d 5095 . . . . 5  |-  ( f  =  K  ->  (
( dist `  f )  |`  ( ( Base `  f
)  X.  ( Base `  f ) ) )  =  ( ( dist `  K )  |`  ( X  X.  X ) ) )
10 isms.d . . . . 5  |-  D  =  ( ( dist `  K
)  |`  ( X  X.  X ) )
119, 10syl6eqr 2461 . . . 4  |-  ( f  =  K  ->  (
( dist `  f )  |`  ( ( Base `  f
)  X.  ( Base `  f ) ) )  =  D )
1211fveq2d 5853 . . 3  |-  ( f  =  K  ->  ( MetOpen
`  ( ( dist `  f )  |`  (
( Base `  f )  X.  ( Base `  f
) ) ) )  =  ( MetOpen `  D
) )
133, 12eqeq12d 2424 . 2  |-  ( f  =  K  ->  (
( TopOpen `  f )  =  ( MetOpen `  (
( dist `  f )  |`  ( ( Base `  f
)  X.  ( Base `  f ) ) ) )  <->  J  =  ( MetOpen
`  D ) ) )
14 df-xms 21115 . 2  |-  *MetSp  =  { f  e.  TopSp  |  ( TopOpen `  f )  =  ( MetOpen `  (
( dist `  f )  |`  ( ( Base `  f
)  X.  ( Base `  f ) ) ) ) }
1513, 14elrab2 3209 1  |-  ( K  e.  *MetSp  <->  ( K  e.  TopSp  /\  J  =  ( MetOpen `  D )
) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 367    = wceq 1405    e. wcel 1842    X. cxp 4821    |` cres 4825   ` cfv 5569   Basecbs 14841   distcds 14918   TopOpenctopn 15036   MetOpencmopn 18728   TopSpctps 19689   *MetSpcxme 21112
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-rex 2760  df-rab 2763  df-v 3061  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-opab 4454  df-xp 4829  df-res 4835  df-iota 5533  df-fv 5577  df-xms 21115
This theorem is referenced by:  isxms2  21243  xmstopn  21246  xmstps  21248  xmspropd  21268
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