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Theorem isms 18484
 Description: Express the predicate " is a metric space" with underlying set and distance function . (Contributed by NM, 27-Aug-2006.) (Revised by Mario Carneiro, 24-Aug-2015.)
Hypotheses
Ref Expression
isms.j
isms.x
isms.d
Assertion
Ref Expression
isms

Proof of Theorem isms
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 fveq2 5731 . . . . 5
2 fveq2 5731 . . . . . . 7
3 isms.x . . . . . . 7
42, 3syl6eqr 2488 . . . . . 6
54, 4xpeq12d 4906 . . . . 5
61, 5reseq12d 5150 . . . 4
7 isms.d . . . 4
86, 7syl6eqr 2488 . . 3
94fveq2d 5735 . . 3
108, 9eleq12d 2506 . 2
11 df-ms 18356 . 2
1210, 11elrab2 3096 1
 Colors of variables: wff set class Syntax hints:   wb 178   wa 360   wceq 1653   wcel 1726   cxp 4879   cres 4883  cfv 5457  cbs 13474  cds 13543  ctopn 13654  cme 16692  cxme 18352  cmt 18353 This theorem is referenced by:  isms2  18485  msxms  18489  mspropd  18509  setsms  18515  tmsms  18522  imasf1oms  18525  ressms  18561  prdsms  18566 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-xp 4887  df-res 4893  df-iota 5421  df-fv 5465  df-ms 18356
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