Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  iscusp Structured version   Unicode version

Theorem iscusp 20927
 Description: The predicate " is a complete uniform space." (Contributed by Thierry Arnoux, 3-Dec-2017.)
Assertion
Ref Expression
iscusp CUnifSp UnifSp CauFiluUnifSt
Distinct variable group:   ,

Proof of Theorem iscusp
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 fveq2 5872 . . . 4
21fveq2d 5876 . . 3
3 fveq2 5872 . . . . . 6 UnifSt UnifSt
43fveq2d 5876 . . . . 5 CauFiluUnifSt CauFiluUnifSt
54eleq2d 2527 . . . 4 CauFiluUnifSt CauFiluUnifSt
6 fveq2 5872 . . . . . 6
76oveq1d 6311 . . . . 5
87neeq1d 2734 . . . 4
95, 8imbi12d 320 . . 3 CauFiluUnifSt CauFiluUnifSt
102, 9raleqbidv 3068 . 2 CauFiluUnifSt CauFiluUnifSt
11 df-cusp 20926 . 2 CUnifSp UnifSp CauFiluUnifSt
1210, 11elrab2 3259 1 CUnifSp UnifSp CauFiluUnifSt
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wa 369   wceq 1395   wcel 1819   wne 2652  wral 2807  c0 3793  cfv 5594  (class class class)co 6296  cbs 14643  ctopn 14838  cfil 20471   cflim 20560  UnifStcuss 20881  UnifSpcusp 20882  CauFiluccfilu 20914  CUnifSpccusp 20925 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-iota 5557  df-fv 5602  df-ov 6299  df-cusp 20926 This theorem is referenced by:  cuspusp  20928  cuspcvg  20929  iscusp2  20930  cmetcuspOLD  21918  cmetcusp  21919
 Copyright terms: Public domain W3C validator