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Theorem isbnd 26491
 Description: The predicate "is a bounded metric space". (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 12-Sep-2015.)
Assertion
Ref Expression
isbnd
Distinct variable groups:   ,,   ,,

Proof of Theorem isbnd
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elfvex 5760 . 2
2 elfvex 5760 . . 3
4 fveq2 5730 . . . . . 6
5 eqeq1 2444 . . . . . . . 8
65rexbidv 2728 . . . . . . 7
76raleqbi1dv 2914 . . . . . 6
84, 7rabeqbidv 2953 . . . . 5
9 df-bnd 26490 . . . . 5
10 fvex 5744 . . . . . 6
1110rabex 4356 . . . . 5
128, 9, 11fvmpt 5808 . . . 4
1312eleq2d 2505 . . 3
14 fveq2 5730 . . . . . . . 8
1514oveqd 6100 . . . . . . 7
1615eqeq2d 2449 . . . . . 6
1716rexbidv 2728 . . . . 5
1817ralbidv 2727 . . . 4
1918elrab 3094 . . 3
2013, 19syl6bb 254 . 2
211, 3, 20pm5.21nii 344 1
 Colors of variables: wff set class Syntax hints:   wb 178   wa 360   wceq 1653   wcel 1726  wral 2707  wrex 2708  crab 2711  cvv 2958  cfv 5456  (class class class)co 6083  crp 10614  cme 16689  cbl 16690  cbnd 26478 This theorem is referenced by:  bndmet  26492  isbndx  26493  isbnd3  26495  bndss  26497  totbndbnd  26500 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405 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-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  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 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-iota 5420  df-fun 5458  df-fv 5464  df-ov 6086  df-bnd 26490
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