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Theorem iscau 19221
 Description: Express the property " is a Cauchy sequence of metric ." Part of Definition 1.4-3 of [Kreyszig] p. 28. The condition allows us to use objects more general than sequences when convenient; see the comment in df-lm 17285. (Contributed by NM, 7-Dec-2006.) (Revised by Mario Carneiro, 14-Nov-2013.)
Assertion
Ref Expression
iscau
Distinct variable groups:   ,,   ,,   ,,

Proof of Theorem iscau
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 caufval 19220 . . 3
21eleq2d 2502 . 2
3 reseq1 5132 . . . . . 6
4 eqidd 2436 . . . . . 6
5 fveq1 5719 . . . . . . 7
65oveq1d 6088 . . . . . 6
73, 4, 6feq123d 5575 . . . . 5
87rexbidv 2718 . . . 4
98ralbidv 2717 . . 3
109elrab 3084 . 2
112, 10syl6bb 253 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359   wceq 1652   wcel 1725  wral 2697  wrex 2698  crab 2701   cres 4872  wf 5442  cfv 5446  (class class class)co 6073   cpm 7011  cc 8980  cz 10274  cuz 10480  crp 10604  cxmt 16678  cbl 16680  cca 19198 This theorem is referenced by:  iscau2  19222  caufpm  19227  lmcau  19257 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-map 7012  df-xr 9116  df-xmet 16687  df-cau 19201
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